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MATHSBOOKS - MATHEMATICS R.D. SHARMA

TRIANGLES

1. Fill in the blanks using the correct word given in brackets: (a) All circles are ................... (congruent,

similar) (b) All squares are ................... (similar, congruent) (c) All ....................... triangles are similar

(isosceles, equilaterals):

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2. Fill in the blanks using the correct word given in brackets: Two triangles are similar, if their corresponding

angles are ...... (proportional, equal) Two triangles are similar, if their corresponding sides are .......

(proportional, equal) (iii) Two polygons of the same number of sides are similar, if (a) their corresponding

angles are and (b) their corresponding sides are ........ (equal, proportional).

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3. Write the truth value (T/F) of each of the following statements:(1) Any two similar figures are congruent.(2)

Any two congruent figures are similar. (3) Two polygons are similar, if their corresponding sides are

proportional.

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4. Write the truth value (T/F) of each of the following statements: (1)Two polygons are similar if their

corresponding angles are proportional. (2)Two triangles are similar if their corresponding sides are

proportional. (3) Two triangles are similar if their corresponding angles are proportional.

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5. In a given ` /_\ ABC ,D E||B C` and `(A D)/(D B)=3/5dot` If `A C=5. 6` , find `A E` .

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6. In Figure, `LM||AB` If `A L=x-3,\ \ A C=2x ,\ \ B M=x-2` and `B C=2x+3` , find the value of `x` .

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7. In Fig. if `S T || Q R` . Find `P S`

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8. In Fig. (i) and (ii), `P Q || B C` . Find `Q C` in (i) and `A Q` in (ii) (FIGURE)

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9. In Fig. if `E F|| D C|| A Bdot` Prove that `(A E)/(E D)=(B F)/(F C)` . (FIGURE)

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10. In Fig. if `E F|| D C|| A Bdot` Prove that `(A E)/(E D)=(B F)/(F C)` . (FIGURE)

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11. In Fig. if `P Q|| B C` and `P R|| C D` . Prove that (i) `(A R)/(A D)=(A Q)/(A B)` (ii) `(Q B)/(A Q)=(D R)/(A

R)` .

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12. Any point `X` inside ` D E F` is joined to its vertices. From a point `P` in `D X ,\ P Q` is drawn parallel to

`D E` meeting `X E` at `Q` and `Q R` is drawn parallel to `E F` meeting `X F` in `R` . Prove that `P R || D F` .

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13. In Fig. 4.31, if `(A D)/(D C)=(B E)/(E C)` and `/_C D E=/_C E D` , prove that ` C A B` is isosceles.

(FIGURE)

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14. In Fig. 4.32, `D E|| A Q` and `D F|| A R` . Prove that `E F || Q R` . (FIGURE)

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15. In Fig. 4.33, `(P S)/(S Q)=(P T)/(T R)` and `/_P S T=/_P R Q` . Prove that ` P Q R` is an isosceles

triangle. (FIGURE)

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16. In Fig. 4.34, `A ,\ B` and `C` are points on `O P ,\ O Q` and `O R` respectively such that `A B P Q` and

`B C Q R` . Show that `A C P R` . (FIGURE)

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17. In a ` A B C` , `D` and `E` are points on the sides `A B` and `A C` respectively such that `D E || B C` If `A

D=6\ c m` , `D B=9\ c m` and `A E=8\ c m` , find `A C` .

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18. In a ` A B C` , `D` and `E` are points on the sides `A B` and `A C` respectively such that `D E// B C` If `A

D=4` , `A E=8` , `D B=x-4` , and `E C=3x-19` , find `x` .

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19. In a ` A B C` , `D` and `E` are points on the sides `A B` and `A C` respectively such that `D E// B C` If `A

D=2\ c m` , `A B=6\ c m` and `A C=9\ c m` , find `A E` . .

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20. In a ` A B C` , `D` and `E` are points on the sides `A B` and `A C` respectively such that `D E B C` If `A

D=8x-7,\ \ D B=5x-3,\ \ A E=4x-3` and `E C=(3x-1)` , find the value of `x` . .

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21. In a ` A B C ,\ \ D\ a n d\ E` are points on the sides `A B` and `A C` respectively. For each of the

following cases show that `D E// B C` : `A B=12\ c m ,\ \ A D=8\ c m ,\ \ A E=12\ c m\ a n d\ A C=18\ c m` .

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22. In a ` A B C ,\ \ D\ a n d\ E` are points on the sides `A B` and `A C` respectively. For each of the

following cases show that `D E// B C` : `A B=10. 8 c m ,\ \ B D=4. 5\ c m ,\ \ A C=4. 8 c m\ a n d\ A E=2. 8 c

m` .

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23. In a ` A B C` , `P` and `Q` are point on the sides `A B` and `A C` respectively, such that `P Q// B C` . If

`A P=2. 4 c m` , `A Q=2c m` , `Q C=3c m` and `B C=6c m` , find `A B` and `P Q` .

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24. In a ` A B C` , `D` and `E` are points on `A B` and `A C` respectively such that `D E||B C` . If `A D=2. 4 c

m ,\ \ A E=3. 2 c m ,\ \ D E=2c m` and `B C=5c m` , find `B D` and `C E` .

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25. In Fig. state if `P Q||E F`

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26. `M` and `N` are points on the sides `P Q` and `P R` respectively of a ` P Q R` . For each of the following

cases, state whether `MN||QR` : `(i) P M=4c m ,\ \ Q M=4. 5 c m ,\ \ P N=4c m ,\ \ N R=4. 5 c m` `(ii) P Q=1.

28 c m ,\ \ P R=2. 56 c m ,\ \ P M=0. 16 c m ,` `P N=0. 32 c m`

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27. If `D` and `E` are points on sides `A B` and `A C` respectively of a triangle ` A B C` such that `D E || B C`

and `B D=C E` . Prove that ` A B C` is isosceles.

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28. In Fig. 4.41, `A D` is the bisector of `/_A` . If `B D=4c m` , `D C=3c m` and `A B=6c m` , determine `A C`

. (FIGURE)

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29. In Fig. 4.42, `A D` is the bisector of `/_B A C` . If `A B=10 c m` . `A C=14 c m` and `B C=6c m` , find `B

D` and `D C` . (FIGURE)

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30. If the diagonal `B D` of a quadrilateral `A B C D` bisects both `/_B` and `/_D` , show that `(A B)/(B C)=(A

D)/(C D)` .

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31. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

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32. In triangle ` A B C` , if `A D` is the bisector of `/_A` , prove that: `(A r e a( A B D))/(A r e a( A C D))=(A

B)/(A C)`

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33. `B O` and `C O` are respectively the bisectors of `/_B` and `/_C` of ` A B C` . `A O` produced meets `B

C` at `P` . Show that: `(A B)/(B P)=(A O)/(O P)` (ii) `(A C)/(C P)=(A O)/(O P)` (iii) `(A B)/(A C)=(B P)/(P C)`

(iv) `A P` is the bisector of `/_B A C` .

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34. In a `/_\ A B C` , `A D` is the bisector of `/_A` , meeting side `B C` at `D` . `(i) If B D=2. 5 c m ,\ \ A B=5c

m` and `A C=4. 2 c m` , find `D C` . `(ii) If B D=2c m ,\ \ A B=5c m` and `D C=3c m ,` find `A C` . `(iii) If A

B=3. 5 c m` , `A C=4. 2 c m` and `D C=2. 8 c m` , find `B D` .

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35. In a `/_\ A B C` , `A D` is the bisector of `/_A` , meeting side `B C` at `D` . `(i) If AB=10 cm` , `AC=14 cm`

and `BC=6cm` , find `B D` and `D C` `(ii) If A C=4. 2 c m` , `D C=6c m` and `B C=10 c m` , find `A B` . `(iii) If

A B=5. 6 c m` , `A C=6c m` and `D C=3c m` , find `B C` .

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36. In a ` A B C` , `A D` is the bisector of `/_A` , meeting side `B C` at `D` . If `A D=5. 6 c m` , `B C=6c m`

and `B D=3. 2 c m` , find `A C` . (ii) If `A B=10 c m` , `A C=6c m` and `B C=12 c m` , find `B D` and `D C` .

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37. In Fig. 4.57, `A E` is the bisector of the exterior `/_C A D` meeting `B C` produced in `E` . If `A B=10 c

m` , `A C=6c m` and `B C=12 c m` , find `C E` . (FIGURE)

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38. In Fig. 4.58, ` A B C` is a triangle such that `(A B)/(A C)=(B D)/(D C),\ \ /_B=70o,\ \ /_C=50o` . Find `/_B

A D` . (FIGURE)

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39. In `/_\ A B C` (Figure), if `/_1=/_2` , prove that `(A B)/(A C)=(B D)/(D C)` .

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40. `D ,\ E` and `F` are the points on sides `B C ,\ C A` and `A B` respectively of ` A B C` such that `A D`

bisects `/_A ,\ \ B E` bisects `/_B` and `C F` bisects `/_C` . If `A B=5c m ,\ \ B C=8c m` and `C A=4c m` ,

determine `A F ,\ C E` and `B D` .

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41. In Fig. 4.60, check whether `A D` is the bisector of `/_A` of ` A B C` in the following: (FIGURE) `A B=5c

m ,\ \ A C=10 c m ,\ \ B D=1. 5 c m` and `C D=3. 5 c m`

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42. In given figure, check whether AD is the bisector of ∠ A of △ ABC in each of the following: (i) AB = 5 cm,

AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm (ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm (iii) AB

= 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm (iv) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm

(v) AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm

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43. In Fig. , `A D` bisects `/_A ,\ \ A B=12 c m ,\ \ A C=20 c m` and `B D=5c m` , determine `C D`

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44. In Fig. `A B||D C` . Find the value of `x`

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45. In Fig. if `A B||C D` , find the value of `x` .

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46. In Figure, if `A B || C D` find the value of `x`.

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47. In Fig. `A B||C D` . If `O A=3x-19 ,\ \ O B=x-4` , `O C=x-3` and `O D=4` , find `x` .

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48. State which pairs of triangles in Figure are similar. Write the similarity criterion used by you for

answering the question and also write the pairs of similar triangles in the symbolic form:

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49. In Fig. 4.87, find `/_F` . (FIGURE)

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50. In Fig. 4.88, ` A C B ~ A P Qdot` If `B C=8c m` , `P Q=4c m` , `B A=6. 5 c m` , `A P=2. 8 c m` , find `C A`

and `A Q` . (FIGURE)

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51. In Fig. 4.89, if ` E D C ~ E B A ,\ \ /_B E C=115o` and `/_E D C=70o` . Find `/_D E C ,\ \ /_D C E ,\ \ /_E

A B ,\ \ /_A E B` and `/_E B A` . (FIGURE)

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52. In Fig. 4.90, if ` P O S ~ R O Q` , prove that `P S Q R` . (FIGURE)

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53. In Fig. 4.90, if `P S|| Q R` , prove that ` P O S ~ R O Q` . (FIGURE)

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54. In Fig. 4.91, `Q A` and `P B` are perpendiculars to `A B` . If `A O=10 c m` , `B O=6c m` and `P B=9c m` .

Find `A Q` . (FIGURE)

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55. In Figure, `/_C A B=90` and `A D_|_B C` . If `A C=75 c m` , `A B=1m` and `B D=1. 25 m` find `A D`.

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56. The perimeters of two similar triangles `A B C` and `P Q R` are respectively 36cm and 24cm. If `P Q=10

c m` , find `A B` .

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57. In Figure if `/_A D E=/_B` show that ` A D E ~ A B C` . If `A D=3. 8 c m` , `A E=3. 6 c m` , `B E=2. 1 c m`

and `B C=4. 2 c m` , find `D E`.

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58. In Figure, `(A O)/(O C)=(B O)/(O D)=1/2` and `A B=5 cm` . Find the value of `D C`.

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59. In Fig. 4.97, if `/_A=/_C` , then prove that ` A O B ~ C O D` . (FIGURE)

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60. In Fig. if `A B_|_B C` and `D E_|_A C` . Prove that ` A B C ~ A E D`

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61. In Fig. 4.99, if `/_P=/_R T S` , prove that ` R P Q ~ R T S` . (FIGURE)

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62. In Fig. , if `(Q T)/(P R)=(Q R)/(Q S)` and `/_1=/_2` . Prove that ` P Q S ~ T Q R` . (FIGURE)

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63. In Fig. 4.101, `A D` and `C E` are two altitudes of ` A B C` . Prove that ` A E F ~ C D F` (ii) ` A B D ~ C B

E`

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64. In Fig. 4.102 (i) and (ii) , if `C D` and `G H` `(D\ a n d\ H` lie on `A B` and `F E` ) are respectively

bisectors of `/_A C B` and `/_E G F` and ` A B C ~ F E G` , prove that (FIGURE) ` D C A ~ H G F` (ii) `(C

D)/(G H)=(A C)/(F G)` (iii) ` D C B ~ H G E`

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65. In Figure, `C D` and `G H` are respectively the medians of ` A B C` and ` E F G` . If ` A B C ~ F E G` ,

prove that `(i) A D C ~ F H G` `(ii) (C D)/(G H)=(A B)/(F E)` `(iii) C D B ~ G H E`

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66. In Figure, if `B D_|_A C` and `C E_|_A B` , prove that ` (i) /_\A E C ~ /_\A D B` `(ii) (C A)/(A B)=(C E)/(D

B)`

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67. `D` is a point on the side `B C` of ` A B C` such that `/_A D C=/_B A C` . Prove that `(C A)/(C D)=(C

B)/(C A)` or, `C A^2=C BxxC D` .

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68. In Fig. 4.106, considering triangles `B E P` and `C P D` , prove that `B PxxP D=E PxxP C` (FIGURE)

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69. `P` and `Q` are points on sides `A B` and `A C` respectively of ` A B C` . If `A P=3c m` , `P B=6c m` , `A

Q=5c m` and `Q C=10 c m` , show that `B C=3\ P Q` .

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70. In Figure, express `x` in terms of `a ,\ b` and `c`.

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71. In Figure, `/_B A C=90dot` and segment `A D_|_B C` . Prove that `A D^2=B DxxD C` .

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72. In `/_\ A B C` , if `A D_|_B C` and `A D^2=B DxxD C` , prove that `/_B A C=90dot` .

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73. In a ` A B C ,\ \ B D` and `C E` are the altitudes. Prove that ` A D B` and ` A E C` are similar. Is ` C D B ~

B E C` ?

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74. In Figure, `A B C D` is a trapezium with `AB||DC` . If `/_\ A E D` is similar to `/_\ B E C` , prove that `A

D=B C` .

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75. If E is a point on side AD produced of a parallelogram ABCD and BE intersects CD at F , Prove that

`/_\ABE~/_\CFB` .

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76. In Figure, `A B C` is a right triangle right angled at `B` and `D` is the foot of the the perpendicular drawn

from `B` on `A C` . If `D M_|_B C` and `D N_|_A B` , prove that: `(i) D M^2=D NxxM C ` ` (ii) D N^2=D MxxA

N`

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77. In Figure, `A D` and `B E` are respectively perpendiculars to `B C` and `A C` . Show that: `(i) /_\ A D C ~

/_\ B E C` `(ii) C AxxC E=C BxxC D` `(iii) /_\ A B C ~ /_\ D E C` `(iv) C DxxA B=C AxxD E`

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78. In Fig. 4.124, `E` is a point on side `C B` produced of an isosceles triangle `A B C` with `A B=A C` . If `A

D_|_B C` and `E F_|_A C` , prove that (i) △ABD∼△ECF (ii) `A BxxE F=A DxxE C` . (FIGURE)

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79. In Fig. 4.125, ` F E C~= G B D` and `/_1=/_2` . Prove that ` A D E A B C` . (FIGURE)

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80. In Fig. 4.126, `(O A)/(O C)=(O D)/(O B)` . Prove that `/_A=/_C` and `/_B=/_D` . (FIGURE)

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81. In Figure, `D E F G` is a square and `/_B A C=90` . Prove that (i) A G F ~ D B G` (ii) A G F ~ E F C` (iii)

D B G ~ E F C` (iv) D E^2=B D x E C`

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82. In ` A B C ,\ \ D E` is parallel to base `B C` , with `D` on `A B` and `E` on `A C` . If `(A D)/(D B)=2/3` , find

`(B C)/(D E)` .

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83. In Figure, if `A D_|_B C` and `(B D)/(D A)=(D A)/(D C)` , prove that ` A B C` is a right triangle.

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84. In Figure, if `/_\ A B E~= /_\A C D` , prove that `/_\ A D E ~ /_\ A B C` .

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85. In Figure, `/_A C B=90dot` and `C D_|_A B` . Prove that `(C B^2)/(C A^2)=(B D)/(A D)`

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86. In Figure, `/_\A C B ~ /_\A P Q` . If `B C=8c m` , `P Q=4c m` , `B A=6. 5 c m` and `A P=2. 8 c m` , find `C

A` and `A Q` .

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87. A vertical stick 10 cm long casts a shadow 8 cm long. At the same time a tower casts a shadow 30m

long. Determine the height of the tower.

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88. In Figure, `AB||QR` . Find the length of `P B` .

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89. In Fig. 4.138, `X Y || B C` . Find the length of `X Y` .

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90. In a right angled triangle with sides `a` and `b` and hypotenuse `c` , the altitude drawn on the

hypotenuse is `x` . Prove that `a b=c x` .

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91. In Figure, `/_A B C=90^o` and `B D_|_A C` . If `B D=8c m` , `A D=4c m` , find `C D` .

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92. In Fig. 4.140, `/_A B C=90o` and `B D_|_A C` . If `A B=5. 7 c m` , `B D=3. 8\ c m` and `C D=5. 4 c m` ,

find `B C` . (FIGURE)

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93. In Figure `D E || B C` such that AE = (1/4) AC If `A B=6 cm`, then 'f i n d' AD`.

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94. In Fig. 4.142, `P A ,\ Q B` and `R C` are each perpendicular to `A C` . Prove that `1/x+1/z=1/y` .

(FIGURE)

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95. In Fig. 4.143, `/_A=/_C E D` , prove that ` C A B∼ C E D` . Also, find the value of `x` . (FIGURE)

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96. The perimeters of two similar triangles are 25cm and 15cm respectively. If one side of first triangle is

9cm, what is the corresponding side of the other triangle?

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97. In ` A B C` and ` D E F` , it is being given that: `A B=5c m ,\ \ B C=4c m` and `C A=4. 2 c m` ; `D E=10 c

m` , `E F=8c m` and `F D=8. 4 c m` . If `A L_|_B C` and `D M_|_E F` , find `A L : D M` .

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98. `D` and `E` are the points on the sides `A B` and `A C` respectively of a triangle ` A B C` such that: `A

D=8c m ,\ \ D B=12 c m ,\ \ A E=6c m` and `C E=9c m` . Prove that `B C=5//2\ D E` .

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99. In ` A B C` , `A L` and `C M` are the perpendiculars from the vertices `A` and `C` to `B C` and `A B`

respectively. If `A L` and `C M` intersect at `O` , prove that: (i) triangle OMA is similar to triangle OLC (ii) `(O

A)/(O C)=(O M)/(O L)`

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100. In Fig. 4.144, we have `A B∥ CD∥ EF` . If `A B=6c m` , `C D=x\ c m` , `E F=10 c m` , `B D=4c m` and

`D E=y\ c m` , calculate the values of `x` and `y` . (FIGURE)

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101. In Fig. 4.145, if `A B_|_B C` , `D C_|_B C` and `D E_|_A C` , prove that ` C E D ∼ A B C` . (FIGURE)

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102. In an isosceles triangle ` A B C`, the base `A B` is produced both the ways to `P` and `Q` respectively,

such that `A PxxB Q=A C^2` . Prove that triangle APC is similar to triangle BCQ.

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103. Diagonals `A C` and `B D` of a trapezium `A BC D` with `A B, D C` intersect each other at the point `O`

. Using similarity criterion for two triangles, show that `(O A)/(O C)=(O B)/(O D)` .

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104. If ` A B C` and ` A M P` are two right triangles, right angled at `B` and `M` respectively such that `/_M A

P=/_B A C` . Prove that `(C A)/(P A)=(B C)/(M P)`

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105. In two similar triangles `A B C` and `P Q R` , if their corresponding altitudes `A D` and `P S` are in the

ratio 4:9, find the ratio of the areas of triangle ` A B C` and triangle ` P Q R` .

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106. If ` ΔA B C ∼ ΔD E F` such that area of ` ΔA B C` is `9\ c m^2` and the area of ` ΔD E F` is `16\ c m^2`

and `B C=2. 1` cm. Find the length of `E F` .

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107. In Fig. 4.166, `P B` and `Q A` are perpendiculars to segment `A Bdot` If `P O=5c m` , `Q O=7c m` and

Area ` P O B=150\ c m^2` find the area of ` Q O A` . (FIGURE)

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108. Prove that the area of the equilateral triangle described on the side of a square is half the area of the

equilateral triangle described on its diagonal.

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109. In Fig. 4.170, `A B C D` is a trapezium in which `A B∥ D C` and `A B=2\ D C` . Determine the ratio of

the areas of ` A O B` and ` C O D` . (FIGURE)

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110. `D ,\ E ,\ F` are the mid-points of the sides `B C ,\ C A` and ` A B` respectively of a `Delta A B C` .

Determine the ratio of the areas of ` D E F` and ` A B C` .

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111. `D` and `E` are points on the sides `A B` and `A C` respectively of a ` A B C` such that `D E B C` and

divides ` A B C` into two parts, equal in area, find `(B D)/(A B)` .

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112. Two isosceles triangles have equal vertical angles and their areas are in the ratio `16 : 25` . Find the

ratio of their corresponding heights.

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113. In Fig. 4.176, `X Y A C` and `X Y` divides triangular region `A B C` into two parts equal in area.

Determine `(A X)/(A B)dot`

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114. Triangles `A B C` and `D E F` are similar. If area `( A B C)=16\ c m^2` , area `( D E F)=25\ c m^2` and

`B C=2. 3 c m ,` find `E Fdot` (ii) If area `( A B C)=9\ c m^2` , area `( D E F)=64\ c m^2` and `D E=5. 1 c m ,`

find `A Bdot` (iii) If `A C=19 c m` and `D F=8c m` , find the ratio of the area of two triangles.

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115. Triangles `A B C` and `D E F` are similar. If area `( A B C)=36\ c m^2` , area `( D E F)=64\ c m^2` and

`D E=6. 2 c m ,` find `A Bdot` (ii) If `A B=1. 2 c m` and `D E=1. 4 c m` , find the ratio of the areas of ` A B C`

and ` D E F` .

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116. In Fig. 4.177, ` A C B ∼ A P Q` . If `B C=10 c m ,` `P Q=5c m ,` `B A=6. 5 c m` and `A P=2. 8 c m` find

`C A` and `A Q` . Also, find the area `( A C B): a r e a\ ( A P Q)` . (FIGURE)

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117. The areas of two similar triangles are `81\ c m^2` and `49\ c m^2` respectively. Find the ratio of their

corresponding heights. What is the ratio of their corresponding medians?

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118. The areas of two similar triangles are `169\ c m^2` and `121\ c m^2` respectively. If the longest side of

the larger triangle is 26cm, find the longest side of the smaller triangle.

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119. Two isosceles triangles have equal vertical angles and their areas are in the ratio `36 : 25` . Find the

ratio of their corresponding heights.

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120. The areas of two similar triangles are `25\ c m^2` and `36\ c m^2` respectively. If the altitude of the first

triangle is 2.4cm, find the corresponding altitude of the other.

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121. The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of

their areas.

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122. `A B C` is a triangle in which `/_A=90^o,\ \ A N_|_B C ,\ \ B C=12 c m` and `A C=5c m` . Find the ratio

of the areas of ` A N C` and ` A B Cdot`

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123. In Fig. 4.178, `D E∥ B C` (FIGURE) (i) If `D E=4c m` , `B C=6c m` and Area `( A D E)=16\ c m^2` , find

the area of ` A B C` . (ii) If `D E=4c m ,` `B C=8c m` and Area `( A D E)=25\ c m^2` , find the area of ` A B C`

. (iii) If `D E : B C=3:5` . Calculate the ratio of the areas of ` A D E` and the trapezium `B C E D` .

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124. In ` A B C ,\ D` and `E` are the mid-points of `A B` and `A C` respectively. Find the ratio of the areas of

` A D E` and ` A B C` .

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125. In Figure, `/_\ A B C` and `/_\ D B C` are on the same base `B C` . If `A D` and `B C` intersect at `O` ,

prove that `(A r e a\ ( A B C))/(A r e a\ ( D B C))=(A O)/(D O)`.

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126. In ` A B C` , `P` divides the side `A B` such that `A P : P B=1:2` . `Q` is a point in `A C` such that `P Q

B C` . Find the ratio of the areas of ` A P Q` and trapezium `B P Q C` .

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127. The areas of two similar triangles are `100\ c m^2` and `49\ c m^2` respectively. If the altitude of the

bigger triangle is 5 cm, find the corresponding altitude of the other.

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128. If ` ΔA B C ∼ ΔD E F` such that `A B=5c m` , area `(Δ A B C)=20\ c m^2` and area `(Δ D E F)=45\ c

m^2` , determine `D E` .

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129. In ` A B C` , `P Q` is a line segment intersecting `A B` at `P` and `A C` at `Q` such that `P Q B C` and

`P Q` divides ` A B C` into two parts equal in area. Find `(B P)/(A B)` .

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130. The areas of two similar triangles `A B C` and `P Q R` are in the ratio `9: 16` . If `B C=4. 5 c m` , find

the length of `Q R` .

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131. `A B C` is a triangle and `P Q` is a straight line meeting `A B` in `P` and `A C` in `Q` . If `A P=1c m` , `P

B=3c m` , `A Q=1. 5 c m` , `Q C=4. 5 cm` , prove that area of `Δ A P Q` is one-sixteenth of the area of ` ΔA

B C` .

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132. If `D` is a point on the side `A B` of `/_\ A B C` such that `A D : D B=3:2` and `E` is a point on `B C`

such that `DE||AC` . Find the ratio of areas of `/_\ A B C` and `/_\ B D E` .

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133. In Figure, ` A B C` is an obtuse triangle, obtuse angled at `B` . If `A D_|_C B` , prove that `A C^2=A

B^2+B C^2+2\ B CxxB D` .

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134. In Figure, `/_B` of ` A B C` is an acute angle and `A D_|_B C` , prove that `A C^2=A B^2+B C^2-2\ B

CxxB D`

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135. A right triangle has hypotenuse of length `p\ c m` and one side of length `q\ c m` . If `p-q=1` , find the

length of the third side of the triangle.

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136. The sides of certain triangles are given below. Determine which of them are right triangles: (i) `a=6c m`

, `b=8c m` and `c=10 c m` (ii) `a=5c m` , `b=8c m` and `c=11 c m` .

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137. A man goes 10m due east and then 24 m due north. Find the distance from the starting point.

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138. A ladder is placed in such a way that its foot is at a distance of 5m from a wall and its tip reaches a

window 12 m above the ground. Determine the length of the ladder.

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139. A ladder 25m long reaches a window of a building 20m above the ground. Determine the distance of

the foot of the ladder from the building.

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140. The hypotenuse of a right triangle is 6 m more than the twice of the shortest side. If the third side is 2 m

less than the hypotenuse, find the sides of the triangle.

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141. In Fig. 4.192, `A B C` is a right triangle right-angled at `B` . `A D` and `C E` are the two medians drawn

from `A` and `C` respectively. If `A C=5c m` and `A D=(3sqrt(5))/2c m` , find the length of `C E` . (FIGURE)

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142. The perpendicular `A D` on the base `B C` of a triangle ` A B C` intersects `B C` at `D` so that `D B=3C

D` . Prove that `2\ A B^2=2\ A C^2+B C^2` .

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143. `A B C` is an isosceles right triangle right-angled at `C` . Prove that `A B^2=2A C^2` .

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144. `A B C` is a triangle in which `A B=A C` and `D` is any point in `B C` . Prove that `A B^2-A D^2=B D C

D` .

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145. In ` A B C` , `A D` is perpendicular to `B C` . Prove that: `A B^2+C D^2=A C^2+B D^2` (ii) `A B^2-B

D^2=A C^2-C D^2`

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146. From a point `O` in the interior of a ` A B C` , perpendiculars `O D ,\ O E` and `O F` are drawn to the

sides `B C ,\ C A` and `A B` respectively. Prove that: (i) `A F^2+B D^2+C E^2=O A^2+O B^2+O C^2-O D^2-

O E^2-O F^2` (ii) `A F^2+B D^2+C E^2=A E^2+C D^2+B F^2`

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147. A point `O` in the interior of a rectangle `A B C D` is joined with each of the vertices `A ,\ B ,\ C` and `D`

. Prove that `O B^2+O D^2=O C^2+O A^2`

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148. `A B C D` is a rhombus. Prove that `A B^2+B C^2+C D^2+D A^2=A C^2+B D^2`

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149. In a triangle `A B C ,\ A C > A B` , `D` is the mid-point of `B C` and `A E_|_B C` . Prove that: (i) `A

B^2=A D^2-B C* D E+1/4B C^2`

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150. In a triangle `A B C ,\ A C > A B` , `D` is the mid-point of `B C` and `A E_|_B C` . Prove that: (i) `A

B^2=A D^2-B C . D E+1/4B C^2 ` (ii) ` A B^2+A C^2=2\ A D^2+1/2B C^2`

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151. In an equilateral triangle `A B C` the side `B C` is trisected at `D` . Prove that `9\ A D^2=7\ A B^2`

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152. In a ` triangle A B C ,\ \ A D_|_B C` and `A D^2=B DxxC Ddot` Prove that ` A B C` is a right triangle.

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153. `P` and `Q` are points on the sides `C A` and `C B` respectively of ` A B C`, right-angled at `C`. Prove

that `A Q^2+B P^2=A B^2+P Q^2`.

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154. `A B C` is an isosceles triangle with `A C=B C` . If `A B^2=2\ A C^2` , prove that ` A B C` is right

triangle.

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155. In ` P Q R ,\ \ Q M_|_P R` and `P R^2-P Q^2=Q R^2` . Prove that `Q M^2=P MxxM R`

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156. Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares

of its sides.

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157. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the triangle is a right-angle

triangle.

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158. The sides of certain triangles are given below. Determine which of them are right triangles. `(i) a=7c m`

, `b=24 c m` and `c=25 c m` (ii) `a=9c m` , `b=16 c m` and `c=18 c m` (iii) `a=1. 6 c m` , `b=3. 8 c m` and

`c=4c m` (iv) `a=8c m` , `b=10 c m` and `c=6c m`

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159. A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

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160. A ladder 17m long reaches a window of a building 15m above the ground. Find the distance of the foot

of the ladder from the building.

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161. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12m,

find the distance between their tops.

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162. In an isosceles triangle `A B C ,\ A B=A C=25 c m ,\ \ B C=14 c m` . Calculate the altitude from `A` on

`B C` .

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163. The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the

ladder is shifted in such a way that its foot is 8m away from the wall, to what height does its tip reach?

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164. Two poles of height 9m and 14m stand on a plane ground. If the distance between their feet is 12m,

find the distance between their tops.

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165. Using Pythagoras theorem determine the length of `A D` in terms of `b` and `c` shown in Figure.

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166. A triangle has sides 5cm, 12cm and 13cm. Find the length to one decimal place, of the perpendicular

from the opposite vertex to the side whose length is 13cm.

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167. `A B C D` is a square. `F` is the mid-point of `A Bdot` `B E` is one third of `B C` . If the area of ` F B

E=108\ c m^2` , find the length of `A C` .

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168. In an isosceles triangle `A B C` , if `A B=A C=13` cm and the altitude from `A` on `B C` is 5 cm, find `B

C` .

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169. In a ` A B C ,\ \ A B=B C=C A=2a` and `A D_|_B C` . Prove that `A D=asqrt(3)` (ii) Area `( A B

C)=sqrt(3)\ a^2`

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170. The lengths of the diagonals of a rhombus are 24cm and 10cm. Find each side of the rhombus.

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171. Each side of a rhombus is 10cm. If one of its diagonals is 16cm find the length of the other diagonal.

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172. In an acute angled triangle , express a median in term of its sides.

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173. Calculate the height of an equilateral triangle each of whose sides measures 12cm.

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174. In right-angled triangle `A B C` in which `/_C=90^@` , if `D` is the mid-point of `B C` , prove that `A

B^2=4\ A D^2-3\ A C^2` .

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175. In Fig. 4.220, `D` is the mid-point of side `B C` and `A E_|_B C` . If `B C=a ,\ \ A C=b ,\ \ A B=c ,\ \ E

D=x ,\ \ A D=p` and `A E=p` and `A E=h` , prove that: (FIGURE) `b^2=p^2+a x+(a^2)/4` (ii) `c^2=p^2-a x+

(a^2)/4` (iii) `b^2+c^2=2\ p^2+(a^2)/2`

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176. In Fig. 4.221, `/_B<90o` and segment `A D_|_B C` , show that `b^2=h^2+a^2+x^2-2\ a x` (ii)

`b^2=a^2+c^2-2\ a x` (FIGURE)

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177. In ` A B C` , `/_A` is obtuse, `P B_|_A C` and `Q C_|_A B` . Prove that: `A BxxA Q=A CxxA P` (ii) `B

C^2=(A CxxC P+A BxxB Q)`

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178. In a right ` A B C` right-angled at `C` , if `D` is the mid-point of `B C` , prove that `B C^2=4\ (A D^2-A

C^2)` .

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179. In a quadrilateral `A B C D` , `/_B=90^o` , `A D^2=A B^2+B C^2+C D^2` , prove that `/_A C D=90^o`

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180. In an equilateral ` A B C` , `A D_|_B C` , prove that `A D^2=3\ B D^2`

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181. ` A B D` is a right triangle right-angled at `A` and `A C_|_B D` . Show that `A B^2=B CdotB D` (ii) `A

C^2=B CdotD C` (iii) `A D^2=B DdotC D` (iv) `(A B^2)/(A C^2)=(B D)/(D C)`

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182. A guy wire attached to a vertical pole of height 18m is 24m long and has a stake attached to the other

end. How far from the base of the pole should the stake be driven so that the wire will be taut?

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183. An aeroplane leaves an airport and flies due north at a speed of 1000km/hr. At the same time, another

aeroplane leaves the same airport and flies due west at a speed of 1200km/hr. How far apart will be the two

planes after `1 1/2` hours?

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184. In each of the figures [4.222 (i)-(iv)] given below, a line segment is drawn parallel to one side of the

triangle and the lengths of certain line-segments are marked. Find the value of `x` in each of the following:

(FIGURE)

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185. In ` A B C` , points `P` and `Q` are on `C A` and `C B` , respectively such that `C A=16 c m` , `C P=10

c m` , `C B=30 c m` and `C Q=25 c m` . Is `P Q ||A B` ?

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186. In Figure, `D E || C B` . Determine `A C` and `A E` .

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187. In Figure, given that `/_\ A B C ~ /_\ P Q R` and `quad\ ABCD ~ quad\ PQRS` . Determine the values of

`x ,\ y ,\ z` in each case.

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188. In ` A B C` , `P` and `Q` are points on sides `A B` and `A C` respectively such that `P Q||B C` . If `A

P=4c m ,\ \ P B=6c m` and `P Q=3c m` , determine `B C` .

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189. In each of the following figures, you find two triangles. Indicate whether the triangles are similar. Give

reasons in support of your answer. (FIGURES)

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190. In `/_\ P Q R` , `M` and `N` are points on sides `P Q` and `P R` respectively such that `P M=15 c m`

and `N R=8c m` . If `P Q=25 c m` and `P R=20 c m` state whether `M N || Q R` .

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191. In ` A B C` , `P` and `Q` are points on sides `A B` and `A C` respectively such that `P Q ||B C` . If `A

P=3c m` , `P B=5c m` and `A C=8c m` , find `A Q` .

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192. In Figure, `/_\ A M B ~ /_\ C M D ;` determine `M D` in terms of `x ,\ y` and `z` .

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193. In ` A B C` , the bisector of `/_A` intersects `B C` in `D` . If `A B=18 c m` , `A C=15 c m` and `B C=22 c

m` , find `B D`

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194. In Figure, `l || m` Name three pairs of similar triangles with proper correspondence; write similarties.

Prove that `(A B)/(P Q)=(A C)/(P R)=(B C)/(R Q)`

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195. In Figure, `A B || D C` Prove that `(i)/_\ D M U ~ /_\ B M V` `(ii) D MxxB V=B MxxD U`

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196. `A B C D` is a trapezium in which `A B D C` . `P` and `Q` are points on sides `A D` and `B C` such that

`P Q A B` . If `P D=18` , `B Q=35` and `Q C=15` , find `A D` .

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197. In ` A B C ,\ D` and `E` are points on sides `A B` and `A C` respectively such that `A DxxE C=A ExxD

B` . Prove that `D E B C` .

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198. `A B C D` is a trapezium having `A B D C` . Prove that `O` , the point of intersection of diagonals,

divides the two diagonals in the same ratio. Also prove that `(a r( O C D))/(a r( O A B))=1/9` , if `A B=3\ C D`

.

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199. Corresponding sides of two triangles are in the ratio `2\ :3` . If the area of the smaller triangle is `48\ c

m^2` , determine the area of the larger triangle.

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200. The areas of two similar triangles are `36\ c m^2` and `100\ c m^2` . If the length of a side of the

smaller triangle in 3 cm, find the length of the corresponding side of the larger triangle.

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201. Corresponding sides of two similar triangles are in the ratio `1:3` . If the area of the smaller triangle in

`40\ c m^2` , find the area of the larger triangle.

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202. In ` A B C` , `A D` and `B E` are altitudes. Prove that `(a r( D E C))/(a r( A B C))=(D C^2)/(A C^2)`

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203. The diagonals of quadrilateral `A B C D` intersect at `O` . Prove that `(a r( A C B))/(a r( A C D))=(B

O)/(D O)`

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204. In Figure, each of `P A ,\ Q B ,\ R C` and `S D` is perpendicular to `l` . If `A B=6c m` , `B C=9c m` , `C

D=12 c m` and `P S=36 c m` , then determine `P Q ,\ Q R` and `R S` .

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205. In ` A B C` , ray `A D` bisects `/_A` and intersects `B C` in `D` . If `B C=a` , `A C=b` and `A B=c` , prove

that `B D=(a c)/(b+c)` (ii) D C = `(a b)/(b+c)`

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206. In each of the figures given below, an altitude is drawn to the hypotenuse by a right-angled triangle.

The length of different line-segments are marked in each figure. Determine `x ,\ y ,\ z` in each case.

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207. There is a staircase as shown in Figure, connecting points `A` and `B` . Measurements of steps are

marked in the figure. Find the straight line distance between `A` and `B` .

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208. In ` A B C ,\ \ /_A=60o` . Prove that `B C^2=A B^2+A C^2-A B.A C .

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209. In ` A B C` , `/_C` is an obtuse angle. `A D_|_B C` and `A B^2=A C^2+3\ B C^2dot` Prove that `B C=C

D` .

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210. A point `D` is on the side `B C` of an equilateral triangle `A B C` such that `D C=1/4\ B C` . Prove that

`A D^2=13\ C D^2` .

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211. In ` A B C` , if `B D_|_A C` and `B C^2=2\ A C C D` , then prove that `A B=A C` .

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212. In a quadrilateral `A B C D` , given that `/_A+/_D=90o` . Prove that `A C^2+B D^2=A D^2+B C^2` .

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213. In `/_\ A B C` , given that `A B=A C` and `B D_|_A C` . Prove that `B C^2=2\ A C * C D`

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214. `A B C D` is a rectangle. Points `M` and `N` are on `B D` such that `A M_|_B D` and `C N_|_B D` .

Prove that `B M^2+B N^2=D M^2+D N^2` .

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215. In ` A B C` , `A D` is a median. Prove that `A B^2+A C^2=2\ A D^2+2\ D C^2` .

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216. In a ` A B C` , `/_A B C=135o` . Prove that `A C^2=A B^2+B C^2+4\ a r( A B C)`

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217. In a quadrilateral `A B C D ,\ \ /_B=90o` . If `A D^2=A B^2+B C^2+C D^2` then prove that `/_A C

D=90o` .

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218. In a triangle `A B C` , `N` is a point on `A C` such that `B N_|_A C` . If `B N^2=A N N C` , prove that

`/_B=90o` .

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219. Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and

the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of

the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have

out? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly

from her after 12 seconds?

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220. State basic proportionality theorem and its converse.

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221. In the adjoining figure, find `A C` . (FIGURE)

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222. In the adjoining figure, if `A D` is the bisector of `/_A` if BD=4cm ,DC=3cm and AB=6cm , what is `A C`

?

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223. State AA similarity criterion.

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224. State `S S S` similarity criterion.

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225. State `S A S` similarity criterion.

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226. In the adjoining figure, `D E` is parallel to `B C` and `A D=1c m`, `B D=2c m`. What is the ratio of the

area of `/_\ A B C` to the area of `/_\ A D E`?

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227. In the figure given below `D E || B C` . If `A D=2. 4 c m` , `D B=3. 6 c m` and `A C=5c m` , Find `A E` .

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228. If the areas of two similar triangles `A B C` and `P Q R` are in the ratio `9: 16` and `B C=4. 5 c m` ,

what is the length of `Q R` ?

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229. The areas of two similar triangles are `169\ c m^2` and `121\ c m^2` respectively. If the longest side of

the larger triangle is 26cm, what is the length of the longest side of the smaller triangle?

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230. If `/_\A B C` and `/_\D E F` are similar triangles such that `/_A=57^0` and `/_E=73^0` , what is the

measure of `/_C` ?

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231. If the altitude of two similar triangles are in the ratio `2:3` , what is the ratio of their areas?

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232. If ` A B C` and `D E F` are two triangles such that `(A B)/(D E)=(B C)/(E F)=(C A)/(F D)=3/4` , then write

Area `( A B C): A r e a( D E F)` .

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233. If ` /_\A B C` and ` /_\D E F` are similar triangles such that `A B=3c m` , `B C=2c m` `C A=2. 5 c m` and

`E F=4c m` , find the perimeter of ` D E F` .

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234. State Pythagoras theorem and its converse.

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235. The lengths of the diagonals of a rhombus are 30cm and 40cm. Find the side of the rhombus.

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236. In Figure, `P Q || B C` and `A P : P B=1:2` . Find `(a r e a( A P Q))/(a r e a( A B C))` .

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237. In Fig. 4.237, `L M=L N=46o` . Express `x` in terms of `a ,\ b` and `c` where `a ,\ b ,\ c` are lengths of `L

M ,\ M N` and `N K` respectively. (FIGURE)

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238. In Figure, `S` and `T` are points on the sides `P Q` and `P R` respectively of `/_\ P Q R` such that `P

T=2c m` , `T R=4c m` and `S T` is parallel to `Q R` . Find the ratio of the areas of ` /_\P S T` and ` /_\P Q R`

.

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239. In Figure, `/_\ A H K` is similar to ` /_\ A B C` . If `A K=10 c m` , `B C=3. 5 c m` and `H K=7c m` , find `A

C` .

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240. In Figure, `D E || B C` in `/_\ A B C` such that `B C=8c m` , `A B=6c m` and `D A=1. 5 c m` . Find `D E`

.

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241. In Fig. 4.241, `D E B C` and `A D=1/2B D` . If `B C=4. 5 c m` , find `D E` . (FIGURE)

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242. A vertical stick 20 m long casts a shadow 10m long on the ground. At the same time, a tower casts a

shadow 50m long on the ground. The height of the tower is (a) 100m (b) 120m (c) 25m (d) 200m

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243. Sides of two similar triangles are in the ratio `4:9` . Areas of these triangles are in the ratio. `2:3` (b)

`4:9` (c) `81 : 16` (d) `16 : 81`

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244. The areas of two similar triangles are `9\ c m^2` and `16\ c m^2` respectively . The ratio of their

corresponding sides is

A. (a)`3:4`

B. (b) `4:3`

C. (c) `2:3`

D. (d) `4:5`

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245. The areas of two similar triangles ` A B C` and ` D E F` are `144\ c m^2` and `81\ c m^2` respectively. If

the longest side of triangle ` A B C` be 36 cm, then the longest side of the triangle ` D E F` is

A. (a) 20cm

B. (b) 26cm

C. (c) 27cm

D. (d) 30cm

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246. ` A B C` and `B D E` are two equilateral triangles such that `D` is the mid-point of `B C` . The ratio of

the areas of the triangles `A B C` and `B D E` is `2:1` (b) `1:2` (c) `4:1` (d) `1:4`

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247. Two isosceles triangles have equal angles and their areas are in the ratio `16 : 25` . The ratio of their

corresponding heights is `4:5` (b) `5:4` (c) `3:2` (d) `5:7`

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248. If ` A B C` and ` D E F` are similar triangles such that `2\ A B=D E` and `B C=8c m` , then `E F=`

A. (a) 16cm

B. (b) 12cm

C. (c) 8cm

D. (d) 4cm

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249. If ` A B C` and ` D E F` are two triangles such that `(A B)/(D E)=(B C)/(E F)=(C A)/(F D)=2/5` , then `A r

e a( A B C): A r e a( D E F)=`

A. (a) `2:5`

B. (b) `4: 25`

C. (c) `4: 15`

D. (d) `8: 125`

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250. Triangle `ABC` is such that `A B=3c m` , `B C=2c m` and `C A=2. 5 c m` . If ` D E F ` `similar ``A B C`

and `E F=4c m` , then perimeter of ` D E F` is

A. (a) `7.5 c m`

B. (b) `15 c m`

C. (c) `22.5 c m`

D. (d) `30 c m`

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251. `X Y` is drawn parallel to the base `B C` of ` Delta A B C` cutting `A B` at `X` and `A C` at `Y` . If `A

B=4\ B X` and `Y C=2c m` , then `A Y=` (a) `2c m` (b) `4c m` (c) `6c m` (d) `8c m`

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252. Two poles of height 6m and 11m stand vertically upright on a plane ground. If the distance between

their foot is 12m, the distance between their tops is (a) 12m (b) 14m (c) 13m (d) 11m

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253. In ` A B C` , a line `X Y` parallel to `B C` cuts `A B` at `X` and `A C` at `Y` . If `B Y` bisects `/_X Y C` ,

then (a)`B C=C Y` (b) `B C=B Y` (c) `B C!=C Y` (d) `B C!=B Y`

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254. In `Delta A B C` , `D` and `E` are points on side `A B` and `A C` respectively such that `D E ` is parallel

to `B C\ ` & `A D : D B=3:1` . If `E A=3. 3 c m` , then `A C=` mmmmmmmm (a) `1. 1 c m` (b) `4c m` (c) `4. 4

c m` (d) `5. 5 c m`

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255. In triangles `A B C` and `D E F` , `/_A=/_E=40^0` , `A B : E D=A C : E F` and `/_F=65^0` , then `/_B=`

`(a) 35^0` `(b) 65^0` `(c) 75^0` `(d) 85^0`

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256. If `/_\A B C` and `/_\D E F` are similar triangles such that `/_A=47^0` and `/_E=83^0` , then `/_C=` (a)

`50^0` (b) `60^0` (c) `70^0` (d) `80^0`

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257. If `D ,\ E` and `F` are the mid-points of the sides of a ` /_\A B C` , the ratio of the areas of the triangles

`D E F` and `ABC` is .......

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258. In a ` A B C` , `/_A=90^o` , `A B=5c m` and `A C=12 c m` . If `A D_|_B C` , then `A D=` `(a) (13)/2c m`

(b) `(60)/(13)c m` (c) `(13)/(60)c m` (d) `(2sqrt(15))/3c m`

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259. In an equilateral triangle `A B C ,` if `A D_|_B C` , then (a) `2\ A B^2=3\ A D^2` (b) `4\ A B^2=3\ A D^2`

(c) `3\ A B^2=4\ A D^2` (d) `3\ A B^2=2\ A D^2`

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260. If ` A B C` is an equilateral triangle such that `A D_|_B C` , then `A D^2=` `(a) 3/2D C^2` (b) `2\ D C^2`

(c) `3\ C D^2` (d) `4\ D C^2`

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261. In a ` A B C` , perpendicular `A D` from `A` on `B C` meets `B C` at `D` . If `B D=8c m` , `D C=2c m`

and `A D=4c m` , then ` A B C` is isosceles (b) ` A B C` is equilateral (c) `A C=2\ A B` (d) ` A B C` is right-

angled at `A` .

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262. In a ` Delta A B C` , point `D` is on side `A B` and point `E` is on side `A C` , such that `B C E D` is a

trapezium. If `D E : B C=3:5` , then Area`( A D E) : A r e a( B C E D)=`

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263. In a ` A B C` , `A D` is the bisector of `/_B A C` . If `A B=6c m` , `A C=5c m` and `B D=3c m` , then `D

C=` `11. 3 c m` (b) `2. 5 c m` (c) `3:5c m` (d) None of these

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264. In a ` A B C` , `A D` is the bisector of `/_B A C` . If `A B=8c m` , `B D=6c m` and `D C=3c m` . Find `A

C` `4c m` (b) `6c m` (c) `3c m` (d) `8c m`

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265. `A B C D` is a trapezium such that `B C || A D` and `A D=4c m` . If the diagonals `A C` and `B D`

intersect at `O` such that `(A O)/(O C)=(D O)/(O B)=1/2` , then `B C=` (a) `7c m` (b) `8c m` (c) `9c m` (d) `6c

m`

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266. If `A B C` is an isosceles triangle and `D` is a point on `B C` such that `A D_|_B C` then,

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267. ` A B C` is a right triangle right-angled at `A` and `A D_|_B C` . Then, `(B D)/(D C)=` mm (a) `((A B)/(A

C))^2` (b) `(A B)/(A C)` (c) `((A B)/(A D))^2` (d) `(A B)/(A D)`

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268. If `E` is a point on side `C A` of an equilateral triangle `A B C` such that `B E_|_C A` , then `A B^2+B

C^2+C A^2=` `2\ B E^2` (b) `3\ B E^2` (c) `4\ B E^2` (d) `6\ B E^2`

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269. If in ` A B C` and ` D E F` , `(A B)/(D E)=(B C)/(F D)` , then ` A B C ~ D E F` when `/_A=/_F` (b)

`/_A=/_D` (c) `/_B=/_D` (d) `/_B=/_E`

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270. If in two triangles `A B C` and `D E F` , `(A B)/(D E)=(B C)/(F E)=(C A)/(F D)` , then ` F D E C A B` (b) `

F D E A B C` (c) ` C B A F D E` (d) ` B C A F D E`

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271. ` A B C D E F` , `a r( A B C)=9c m^2` , `a r( D E F)=16 c m^2` . If `B C=2. 1 c m` , then the measure of

`E F` is `2. 8 c m` (b) `4. 2 c m` (c) `2. 5 c m` (d) `4. 1 c m`

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272. The length of the hypotenuse of an isosceles right triangle whose one side is `4sqrt(2)` cm is `12 c m`

(b) `8c m` (c) `8sqrt(2)c m` (d) `12sqrt(2)c m`

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273. A man goes 24m due west and then 7m due north. How far is he from the starting point? (a) 31m

(b) 17m (c) 25m (d) 26m

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274. If ` Delta A B C ~ Delta D E F` such that `B C=3c m` , `E F=4c m` and `a r( A B C)=54 c m^2` , then `a

r( D E F)=`

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275. If ` Delta A B C ~ Delta P Q R` such that `a r( A B C)=4a r( P Q R)dot` If `B C=12 c m` , then `Q R=`

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276. The areas of two similar triangles are `121\ c m^2` and `64\ c m^2` respectively. If the median of the

first triangle is `12. 1\ c m` , then the corresponding median of the other triangle is (a) 11cm (b) 8.8cm (c)

11.1cm (d) 8.1cm

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277. If ` A B C D E F` such that `D E=3c m` , `E F=2c m` , `D F=2. 5 c m` , `B C=4c m` , then perimeter of `

A B C` is (a) 18cm (b) 20cm (c) 12cm (d) 15cm

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278. In an equilateral triangle `A B C` if `A D_|_B C` , then

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279. If ` A B C D E F` such that `A B=9. 1 c m` and `D E=6. 5 c m` . If the perimeter of ` D E F` is 25cm, then

the perimeter of ` A B C` is (a) 36cm (b) 30cm (c) 34cm (d) 35cm

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280. In an isosceles triangle `A B C` if `A C=B C` and `A B^2=2A C^2` , then `/_C=` `30o` (b) `45o` (c) `90o`

(d) `60o`

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281. ` A B C` is an isosceles triangle in which `/_C=90o` . If `A C=6c m` , then `A B=` `6sqrt(2)c m` (b) `6c

m` (c) `2sqrt(6)c m` (d) `4sqrt(2)c m`

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282. If in two triangles `A B C` and `D E F` , `/_A=/_E ,\ \ /_B=/_F` , then which of the following is not true?

`(B C)/(D F)=(A C)/(D E)` (b) `(A B)/(D E)=(B C)/(D F)` (c) `(A B)/(E F)=(A C)/(D E)` (d) `(B C)/(D F)=(A B)/(E

F)`

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283. In an isosceles triangle `A B C` , if `A B=A C=25 c m` and `B C=14 c m` , then the measure of altitude

from `A` on `B C` is (a) 20cm (b) 22cm (c) 18cm (d) 24cm

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284. In Fig. 4.242 the measures of `/_D` and `/_F` are respectively (FIGURE) `50,\ 40` (b) `20,\ 30` (c) `40,\

50` (d) `30,\ 20`

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285. In Figure, the value of `x` for which `D E || A B` (a) 4 (b) 1 (c) 3 (d) 2

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286. In Fig. 4.244, if `/_A D E=/_A B C ,` then `C E=` (FIGURE) (a) 2 (b) 5 (c) `9//2` (d) 3

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287. In Fig. 4.245, `R S D B P Q` . If `C P=P D=11 c m` and `D R=R A=3c m` . Then the values of `x` and

`y` are respectively (FIGURE) `12 ,\ 10` (b) `14 ,\ 6` (c) `10 ,\ 7` (d) `16 ,\ 8`

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288. In Fig. 4.246, if `P B C F` and `D P E F` , then `(A D)/(D E)=` (FIGURE) `3/4` (b) `1/3` (c) `1/4` (d) `2/3`

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289. A chord of a circle of radius 10cm subtends a right angle at the centre. The length of the chord (in cm)

is

A. (a) `5sqrt(2)`

B. (b) `10sqrt(2)`

C. (c) `5/(sqrt(2))`

D. (d) `10sqrt(3)`

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290. In Figure `P Q` is parallel to `M Ndot` if `(K P)/(P M)=4//13` and `K N=20. 4c mdot` Find KQ.

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291. In Figure, `D E || B C ,` if `A D=x ,D B=x-2,A E=x+2` and `E C=x-1,` find the value of `xdot`

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292. Lex X by any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA

meeting CA, BA in M, N respectively; MN meets BC produced in T, prove that `T X^2=T B` x `T C`

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293. ABCD is a parallelogram. P is a point on the side BC DP when produced meets AB produced at L.

Prove that `(D P)/(P L)=(D C)/(B L)` (ii) `(D L)/(D P)=(A L)/(D C)`

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294. In Figure `D E A C` and `D C a pdot` Prove that `(B E)/(E C)=(B C)/(C P)`

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295. In Figure DE||BC and CD||EF. Prove that `A D^2=A B*A F`.

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296. `Da n dE` are respectively the points on the side `A Ba n dA C` of a ` A B C` such that `A B=5. 6 c m ,A

D=1. 4 c m ,A C=7. 2 c m` and `A E=1. 8 c m ,` show that `D E ||B Cdot`

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297. Two triangle ABC and `D B C` lie on the same side of the base `B C` . From a point `P` on BC, PQ||AB

and PR||BD are drawn. They meet `A C` in `Q` and `D C` in `R` respectively. Prove that QR||AD`.

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298. `A B C D` is a quadrilateral; `P , Q , Ra n dS` are the points of trisection of side `A B ,B C ,C Da n dD

A` respectively and are adjacent to `Aa n dC` ; prove that `P Q R S` is parallelogram.

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299. Let `A B C` be a triangle and `D and E` be two points on side `A B` such that `A D=B E`. If `DP || BC`

and `EQ || AC ,` Then prove that `PQ || AB`.

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300. In a ` A B C ,D` and `E` are points on sides `A B a n d A C` respectively such that `B D=C E`. If

`/_B=/_C ,` show that `DE || BC`.

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301. In Figure, `A B C` is a triangle in which `A B=A Cdot` Point `D and E` are points on the sides `AB and

AC` respectively such that `A D=A Edot` Show that the points `B , C , E and D` are concyclic.

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302. The side `B C` of a triangle `A B C` is bisected at `D ; O` is any point in `A D. BO and CO` produced

meet `AC and AB` in `E and F` respectively and `A D` is produced to `X` so that `D` is the mid-point of `O

X`. Prove that `A O : A X=A F : A B` and show that `FE||BC`.

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303. In three line segments `OA ,OB and OC ,` point `L ,M ,N` respectively are so chosen that `LM || AB and

MN || BC` but neither of `L ,M ,N` nor of `A , B , C` are collinear. Show that `LN || AC`.

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304. ABCD is a quadrilateral in which AB=AD . The bisector of BAC and CAD intersect the sides BC

and CD at the points E and F respectively. Prove that EF||BD.

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305. `O` is any point inside a triangle `A B C` . The bisector of `/_A O B ,/_B O C` and `/_C O A` meet the

sides `A B ,B C` and `C A` in point `D ,E and F` respectively. Show that `A D * B E * C F=D B * E C * F A`

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306. `A D` is a median of ` Delta A B C` . The bisector of `/_A D B` and `/_A D C` meet AB and AC in E and

F respectively. Prove that EF||BC

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307. In ` /_\A B C ,D` is the mid-point of `BC and ED` is the bisector of the `/_ADB and EF` is drawn parallel

to `B C` cutting `A C` in `F` . Prove that `/_E D F` is a right angle.

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308. In a quadrilateral `A B C D ,` if bisectors of the `/_A B Ca n d/_A D C` meet on the diagonal `A C` ,

prove that the bisectors of `/_B A Da n d/_B C D` will meet on the diagonal `B Ddot`

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309. In ` A B C` , the bisector of `/_B` meets `A C` at `D`. A line `PQ||AC` meets `A B , BC and BD` at `P ,Q

and R` respectively. Show that (i) `PR*BQ=QR*BP` (ii) `AB*CQ=BC*AP`

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310. The bisectors of the angles B and C of a triangle `A B C` , meet the opposite sides in D and E

respectively. If DE||BC , prove that the triangle is isosceles.

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311. In ` A B C ,/_B=2/_C` and the bisector of `/_B` intersects `A C` at `Ddot` Prove that `(B D)/(D A)=(B

C)/(B A)`

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312. In Figure, `/_B A C=90^0,A D` is its bisector. If `D E_|_A c` , prove that `DE ×(A B+A C)=A B×A C`

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313. Prove that the line segments joints joining the mid-points of the adjacent sides of a quadrilateral from a

parallelogram.

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314. In figure, `P` is the mid-point of `B C , ,Q` is the mid-point of `A P` , such that `B Q` produced meets `A

C` at `Rdot` Prove that `3RA= CA`

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315. A vertical stick 12m long casts a shadow 8m long on the ground. At the same time a tower casts the

shadow 40m long onthe ground. Determine the height of the tower.

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316. The perimeters of two similar triangles are 30cm and 20cm respectively. If one side of the first triangle

is 12cm, determine the corresponding side of the second triangle.

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317. Two triangles `B A Ca n dB D C` , right angled at `Aa n dD` respectively, are drawn on the same base

`B C` and on the same side of `B C` . If `A C` and `D B` intersect at `P ,` prove that `A P*P C=D P*P Bdot`

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318. Two poles of height a metres and `b` metres are `p` metres apart. Prove that the height of the point of

intersection of the lines joining the top of each pole to the foot of the opposite pole is given by `(a b)/(a+b)`

metres.

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319. ABC is a triangle in which `AB = AC and D` is a point on AC such that `BC^2 = AC xx CD`.Prove that

`BD = BC`.

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320. The diagonal `B D` of a parallelogram `A B C D` intersects the segment `A E` at the point `F ,` where E

is any point on the side `B C` . Prove that `D F*E F=F B*F Adot`

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321. Through the mid-point `M` of the side `C D` of a parallelogram `A B C D` , the line `B M` is drawn

intersecting `A C` at `La n dA D` produced at `E` . Prove that `E L=2B Ldot`

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322. If `A B C D` is quadrilateral and `E and F` are the mid-points of `AC and BD` respectively, prove that `

vec A B+ vec A D` +` vec C B` +` vec C D` =4 ` vec E Fdot`

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323. `A B C` is an isosceles triangle with `A B=A C` and `D` is a point on `A C` such that `B C^2=A CXC D`.

Prove that `B D=B C`.

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324. In a triangle ` A B C` , let P and Q be points on AB and AC respectively such that `P Q || B C` . Prove

that the median `A D` bisects `P Q`.

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325. If a perpendicular is drawn from the vertex containing the right angle of a right triangle to the

hypotenuse then prove that the triangle on each side of the perpendicular are similar to each other and to

the original triangle. Also, prove that the square of the perpendicular is equal to the product of the lengths of

the two parts of the hypotenuse.

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326. Prove that the line segments joining the mid-points of the sides of a triangle from four triangles, each of

which is similar to the original triangle.

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327. Through the mid-point `M` of the side `C D` of a parallelogram `A B C D` , the line `B M` is drawn

intersecting `A C` at `La n dA D` produced at `E` . Prove that `E L=2B Ldot`

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328. D is the mid-point of side BC of a triangle ABC.AD is bisected at the point E and BE produced cuts AC

at the point X. Prove that BE:EX=3:1.

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329. `A B C D` is a parallelogram and `A P Q` is a straight line meeting `B C` at `Pa n dD C` produced at

`Qdot` prove that the rectangle obtained by `B Pa n dD Q` is equal to the rectangle contained by `A Ba n dB

Cdot`

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330. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a

diagonal. Show that : (i) `S R` `||` `A C` and `S R=1/2A C` (ii) `P Q = S R` (iii) PQRS is a parallelogram

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331. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is

3.6 m above the ground, find the length of her shadow after 4 seconds.

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332. If triangle ` A B C` is similar to triangle ` D E F` such that `B C=3cm ,E F=4c m` and area of triangle ` A

B C=54c m^2` . Determine the area of triangle ` D E Fdot`

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333. Prove that the area of equilateral triangle described on the side of a square is half the area of the

equilateral triangle described on its diagonal.

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334. Equilateral triangles are drawn on the sides of a right triangle. Show that the area of the triangle on the

hypotenuse is equal to the sum of the areas of triangles on the other two sides.

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335. In the trapezium ABCD,AC and BD intersect at O and also AB=2CD If the area of ` A O B=84 c m^2,`

find the area of ` C O Ddot`

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336. In Figure, `DE||BC and AD : D B=4: 5.` Find `(A r e a( D E F))/(A r e a( C F B))dot`

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337. AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is

constructed. Prove that Area ( triangle ADE ): Area ( triangle ABC )=3:4.

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338. Prove that in any triangle the sum of squares of any two sides is equal to twice the square of half the

third side together with twice the square of the median.

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339. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of

the squares of the medians of the triangle.

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340. A ladder `15 m` long reaches a window which is 9m above the ground on one side of a street. Keeping

its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find

the width of the street.

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341. P and Q are the mid-points of the CA and CD respectively of a triangle ABC, right angled at C. Prove

that: `4AQ^2 = 4AC^2 + BC^2`, `4BP^2 = 4BC^2 + AC^2`, and `4(AQ^2 + BP^2) = 5AB^2`.

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342. ABC is a right triangle right-angled at B. Let D and E be any points on AB and BC respectively. Prove

that `AE^2+ CD^2= AC^2+ DE^2`

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343. In a triangle ` A B C ,` the angles at `B and C` are acute. If `BE and CF` be drawn perpendiculars on

`AC and AB` respectively, prove that `B C^2=A B*B F+A C*C Edot`

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344. In Figure, `A B C` is a right triangle right angled at `B` and points `Da n dE` trisect `B C` . Prove that

`8A E^2=3A C^2+5A D^2dot`

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345. `A B C` is a right triangle right-angled at `Ca n dA C=sqrt(3)B C` . Prove that `/_A B C=60^0dot`

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346. If a perpendicular is drawn from the vertex containing the right angle of a right triangle to the

hypotenuse then prove that the triangle on each side of the perpendicular are similar to each other and to

the original triangle. Also, prove that the square of the perpendicular is equal to the product of the lengths of

the two parts of the hypotenuse.

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347. ABC is an isosceles triangle right-angled at B. Similar triangles ACD and ABE are constructed on side

AC and AB. Find the ratio between the areas of triangle ABE and triangle ACD.

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348. ABC is a right angle triangle right angled at A. A circle is inscribed in it the length of the two

sidescontaining right angle are 6 cm and 8 cm then the radius of the circle is

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349. If `A` be the area of a right triangle and `b` one of the sides containing the right angle, prove that the

length of the altitude on the hypotenuse is `(2A B)/(sqrt(b^4+4A^2))`

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350. Determine whether the triangle having sides `(a-1)c m ,2sqrt(a)c m` and `(a+1)c m` is a right angled

triangle.

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351. In Figure, `D , E` are points on sides `A B` and `A C` respectively of ` /_\A B C ,` such that `a r( B C

E)=a r( B C D)`. Show that `DE||BC`.

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352. If `A B C` is a right triangle right-angled at `Ba n dM ,N` are the mid-points of `A Ba n dB C`

respectively, then `4(A N^2+C M^2)=` (A) `4A C^2` (B) `5A C^2` (C) `5/4A C^2` (D) `6A C^2`

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353. In a right triangle `A B C` right-angled at `B ,` if `P a n d Q` are points on the sides `A Ba n dA C`

respectively, then (a)`A Q^2+C P^2=2(A C^2+P Q^2)` (b)`2(A Q^2+C P^2)=A C^2+P Q^2` (c)`A Q^2+C

P^2=A C^2+P Q^2` (d)`A Q+C P=1/2(A C+P Q)dot`

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354. In an equilateral triangle `A B C` if `A D_|_B C ,` then `A D^2` = (a)`C D^2` (b) `2C D^2` (c) `3C D^2`

(d) `4C D^2`

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