Geometry – Semester One Topics

Geometry – Semester One Topics Chapter 1 – Points, Lines, Plane and Angles 1-2 Points, Lines and Planes 1-3 Segments, Rays, and Distance 1-4 Angles 1-5 Postulates and Theorems Relating Points, Lines and Planes Chapter 2 – Deductive Reasoning 2-1 If-Then Statements; Converses 2-2 Properties of Algebra 2-3 Proving Theorems 2-4 Special Pairs of Angles 2-5 Perpendicular Lines 2-6 Planning a Proof Chapter 3 – Parallel Lines and Planes 3-1 Definitions 3-2 Properties of Parallel Lines 3-3 Proving Lines Parallel 3-4 Angles of a Triangle 3-5 Angles of a Polygon Chapter 4 – Congruent Triangles 4-1 Congruent Figures 4-2 Some Ways to Prove Triangles Congruent 4-3 Using Congruent Triangles 4-4 The Isosceles Triangle Theorems 4-5 Other Methods of Proving Triangles Congruent 4-6 Using More than One Pair of Congruent Triangles 4-7 Medians, Altitudes, and Perpendicular Bisectors Chapter 5 - Quadrilaterals 5-1 Parallelograms 5-2 Ways to Prove that Quadrilaterals are Parallelograms

Geometry – Semester Two Topics Chapter 5 - Quadrilaterals 5-1 Parallelograms 5-2 Ways to Prove that Quadrilaterals are Parallelograms 5-3 Theorems Involving Parallel Lines 5-4 Special Parallelograms 5-5 Trapezoids Chapter 7 – Similar Polygons 6-4 Triangle Inequalities 7-1 Ratio, Proportion and Similarities 7-2 Properties of Proportions 7-3 Similar Polygons 7-4 A Postulate for Similar Triangles 7-5 Theorems for Similar Triangles 7-6 Proportional Lengths Chapter 8 – Right Triangles 8-1 Similarity in Right Triangles (Geometric Mean) 8-2 Pythagorean Theorem 8-3 Converse of the Pythagorean Theorem 8-4 Special Right Triangles Chapter 9 - Circles 9-1 Basic Terms 9-2 Tangents 9-3 Arcs and Central Angles 9-4 Arcs and Cords 9-5 Inscribed Angles 9-6 Other Angles 9-7 Circles and Lengths of Segments Chapter 11 – Areas of Plane Figures 11-1 Areas of Rectangles 11-2 Areas of Parallelograms, Triangles, and Rhombuses 11-3 Areas of Trapezoids 11-4 Areas of Regular Polygons 11-5 Circumference and Area of Circles 11-6 Arc Lengths and Areas of Sectors 11-7 Ratio of Areas Chapter 12 – Areas and Volumes of Solids 12-1 Prisms 12-2 Pyramids 12-3 Cylinders and Cones 12-4 Spheres 12-5 Areas and Volumes of Similar Solids

RIGHT PRISM

REGULAR PYRAMID CYLINDER CONE SPHERE

L.A. 𝑝𝑝ℎ 12𝑝𝑝𝑝𝑝 2𝜋𝜋𝜋𝜋ℎ 𝜋𝜋𝜋𝜋𝑝𝑝 ----------------

T.A. 𝐿𝐿.𝐴𝐴. +2𝐵𝐵 𝐿𝐿.𝐴𝐴. +𝐵𝐵 𝐿𝐿.𝐴𝐴. +2𝐵𝐵 𝐿𝐿.𝐴𝐴. +𝐵𝐵 4𝜋𝜋𝜋𝜋2

V 𝐵𝐵ℎ 13𝐵𝐵ℎ 𝜋𝜋𝜋𝜋2ℎ

13𝜋𝜋𝜋𝜋2ℎ

43𝜋𝜋𝜋𝜋3

GEOMETRY SEMESTER ONE – REVIEW SHEET

PART I: Fill in the Blanks Given: 𝑙𝑙 ∥ 𝑚𝑚, 𝐴𝐴𝐴𝐴�⃖��⃗ ∥ 𝐸𝐸𝐸𝐸�⃖��⃗ , 𝐴𝐴𝐸𝐸�⃖��⃗ ∥ 𝑋𝑋𝑋𝑋�⃖��⃗ , X is the midpoint of 𝐴𝐴𝐸𝐸, Y is the midpoint of 𝐸𝐸𝐸𝐸

1) If ABC∆ is equilateral, then 5m∠ = .

2) If 𝑚𝑚∠6 = 50°, then 14m∠ = .

3) If AB BC≅ , and 𝑚𝑚∠4 = 110°, then 2m∠ = .

4) If 𝑚𝑚∠9 = 130°, then 11m∠ = .

5) If AB AC≅ , and 𝑚𝑚∠5 = 80°, then 2m∠ = .

6) If 𝑚𝑚∠4 = 110°, then 1m∠ = .

7) If 4 5 5m x∠ = + and 11 5 15m x∠ = + , then 10m∠ = .

8) If 8 2 40m x∠ = + and 1 3 20m x∠ = + , then 7m∠ = .

9) If AB BC≅ , 𝑚𝑚∠2 = 30°, then 3m∠ = .

10) If 22BE = , then XE = .

11) If 𝐴𝐴𝐸𝐸��⃗ bisects ABE∠ , 𝑚𝑚∠5 = 40°, then 15m∠ = .

12) If 4 15 2m x∠ = − , 2 3 1m x∠ = − , and 12 2 16m x∠ = + , then x = .

13) 3 13 17 16m m m m∠ + ∠ + ∠ + ∠ = .

14) Find the sum of the measures of the angles of the polygon ABXYC .

15) If 4 20AB x= + and 2 48CE x= + , then x = .

16) If 𝑚𝑚∠3 = 80°, then 15m∠ = .

17) If 𝑚𝑚∠16 = 125°, then 3m∠ = .

18) If 𝑚𝑚∠4 = 125°, and 𝑚𝑚∠2 = 50°, then 12m∠ = .

19) The total number of diagonals than can be drawn in polygon ABXYC is .

20) Draw CX . CX is called a of BEC∆ .

21) If 12 5 2m x∠ = + , 13 6m x∠ = , and 14 6 8m x∠ = + , then 3m∠ = .

22) If 1 7 1m x∠ = − , and 7 9 11m x∠ = − , then 6m∠ = .

23) If 𝑚𝑚∠5 = 88°, then 6m∠ =

24) If XY YE= , and 𝑚𝑚∠15 = 58°, then 18m∠ = .

25) If 𝑚𝑚∠7 = 142° and AB AC≅ , then 2m∠ = .

m

l7

61516

18

17

1413

12

89

54

1

10 11

32 E

X

C

B

A

Y

D

PART II: Always/Sometimes/Never, True/False, and Multiple Choice Complete each statement with the word always, sometimes, or never. 26) When there is a transversal of two lines, the three lines are coplanar. 27) Three lines intersecting in one point are coplanar. 28) Two lines that are not coplanar intersect. 29) Two lines parallel to a third line are parallel to each other. 30) Two lines skew to a third line are skew to each other. 31) Two lines perpendicular to a third line are perpendicular to each other. 32) Two planes parallel to the same line are parallel to each other. 33) Two planes parallel to the same plane are parallel to each other. 34) Lines in two parallel planes are parallel to each other. 35) Two lines parallel to the same plane are parallel to each other. 36) Two lines that do not intersect are parallel. 37) Two skew lines intersect. 38) Two lines parallel to a third line are parallel. 39) If a line is parallel to plane X and also to plane Y , then plane X and plane Y are parallel. 40) Plane X is parallel to plane Y . If plane Z intersects X in line k and Y in line n , then k is

parallel to n . 41) If a triangle is isosceles, then it is equilateral. 42) If a triangle is equilateral, then it is isosceles. 43) If a triangle is scalene, then it is isosceles. 44) If a triangle is obtuse, then it is isosceles. Complete each statement with the word true or false. 45) Three given points are always coplanar.

46) Each interior angle of a regular n gon− has measure ( 2)180n

n

−.

47) If RST RSV∆ ≅ ∆ , then SRT SRV∠ ≅ ∠ . 48) Corresponding parts of similar triangles must be congruent. 49) A point lies on the bisector of ABC∠ if and only if it is equidistant from A and C .

Multiple Choice Problems from Textbook 50) Page 626 # 1 - 11 all 51) Page 627 # 1 – 11 all 52) Page 628 # 1 – 9 all, 11, 12 53) Page 629 # 1 - 14 all 54) Page 630 # 4, 9 PART III: Proofs

55) Given: DG FE≅ , GP EO≅ , GPD∠ and EOF∠ are right angles Prove: GPD EOF∆ ≅ ∆

56) Given: Quad. DEFG is a parallelogram,

GP DF⊥ , EO DF⊥ Prove: GPF EOD∆ ≅ ∆ 57) Given: 𝑗𝑗 ∥ 𝑘𝑘, 4 10∠ ≅ ∠ Prove: 𝑙𝑙 ∥ 𝑚𝑚

58) Given: M is the midpoint of GH G H∠ ≅ ∠ , 1 3∠ ≅ ∠ Prove: 2 4∠ ≅ ∠

59) Given: AC AD≅ , BC BD≅ Prove: CMD∆ is isosceles 60) Given: 1 2∠ ≅ ∠ , 3 4∠ ≅ ∠ Prove: DXC∆ is isosceles

F

E

42

31

P

D

O

G

F

E

42

31

P

D

O

G

k

l m

j

16151413

1211109

87

1 2 3 4

5 6

43

21

J

K

H

M

G

42

31

MG

C

B

D

H

A

4321

X

CD

BA

61) Given: PU SR≅ , 1 2∠ ≅ ∠ , RQ UT≅ Prove: Q T∠ ≅ ∠

62) Given: D and E are midpoints, BAC BCA∠ ≅ ∠

Prove: ADC CEA∆ ≅ ∆

PART IV: Algebraic Problems 63) If 1 5m x∠ = , 2 7 2m x∠ = − , and 11 4m AOB x∠ = + , then x = . 64) If 1 2 28m x∠ = + and 2 3 14m x∠ = − , then 3m∠ = . 65) The measure of the supplement of an angle is four times the measure of its complement. Find the measure of

the angle. 66) If 1∠ and 2∠ are the acute angles of a right triangle, and 1 30m x∠ = + and 2 40m x∠ = + , find the

measure of each angle. Given: 𝑎𝑎 ∥ 𝑏𝑏, 𝑐𝑐 ∥ 𝑑𝑑 67) If 1 11( 1)m x∠ = − and 2 2(4 11)m x∠ = + , then 3m∠ = .

2

1

A

O

B

C

23

1

d

a b

c

1

23 4

5

6

S

T

U

R

Q

P 21

CA

D E

B

Given: 𝑎𝑎 ∥ 𝑏𝑏, 𝑐𝑐 ∥ 𝑑𝑑 68) If 4 2 108m x∠ = + and 26m x x∠ = − , then 5m∠ = . 69) If 1 6 10m x∠ = + , 2 2 4m x∠ = + , and 3 4m x∠ = + , then 4m∠ = . 70) If 1 27 4m x∠ = − , 2 11 4m x∠ = + , and 4 50 48m x∠ = − , then 3m∠ = . 71) The number of diagonals from a single vertex in a regular polygon is 12. Find the measure of each interior angle

of the polygon. 72) If each of 4 interior angles of a convex pentagon has a measure of 105°, find the measure of the fifth interior

angle. 73) Find the total number of diagonals that can be drawn in a regular polygon if each interior angle has a measure of

135°.

4 3 2

1

d

a b

c

1

23 4

5

6

4 3 2

1

74) Polygon ABCDE is regular. Find 2m∠ .

75) ABC∆ is isosceles with AB AC≅ . If 10 8m A x y∠ = − , 2m B x y∠ = + , and 4 5m C x y∠ = − , find the measure of the vertex angle.

76) DEF∆ is equilateral. Find x and y if 10 4 4m D x y∠ = − + and 2 9 10m E x y∠ = + − . Given: Quad. SOPH is a parallelogram. 77) If 12SH = , then OP = . 78) If 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 76°and 𝑚𝑚∠𝑆𝑆𝑆𝑆𝑆𝑆 = 32°, then m OHP∠ = . 79) If 8 30m SOP x∠ = + and 20 10m SHP x∠ = + , then x = . Given: 𝑚𝑚∠𝐸𝐸𝐴𝐴𝐸𝐸 = 30°. Find the indicated measures. 80) m ADF∠ = 81) m FAD∠ = 82) m AMR∠ =

83) m MAR∠ = 84) m AED∠ = 85) m DAR∠ =

E D

C

B

A

2

PH

O

A

S

RMEDF

A

GEOMETRY SEMESTER ONE – REVIEW SHEET - ANSWERS

PART I: Fill in the Blanks 1) 60 2) 50 3) 40 4) 50 5) 50 6) 70 7) 85

8) 100 9) 75 10) 11 11) 70

12) 1710

13) 360 14) 540 15) 14 16) 80 17) 55 18) 75 19) 5

20) median 21) 52 22) 83 23) 88 24) 64 25) 71

PART II: Always/Sometimes/Never, True/False, and Multiple Choice 26) A 27) S 28) N 29) A 30) S 31) S

32) S 33) A 34) S 35) S 36) S 37) N

38) A 39) S 40) A 41) S 42) A 43) N

44) S 45) T 46) T 47) T 48) F 49) F

50) Pg. 626 (1) B (2) D (3) B (4) A (5) A (6) B (7) D (8) C (9) A (10) B (11) A 51) Pg. 627 (1) C (2) A (3) B (4) D (5) D (6) A (7) B (8) B (9) A (10) B (11) A 52) Pg. 628 (1) D (2) A (3) B (4) A (5) D (6) B (7) D (8) C (9) A (11) C (12) A 53) Pg. 629 (1) ASA or AAS (2) AAS (3) ASA or AAS (4) HL (5) SSS (6) AAS (7) SAS

(8) HL (9) D (10) A (11) A (12) C (13) B (14) C 54) Pg. 630 (4) D (9) B PART III: Proofs 55) Statements Reasons

1) DG FE≅ , GP EO≅ GPD∠ and EOF∠ are right angles

1) Given

2) GPD∆ and EOF∆ are right triangles 2) def of rt. triangle 3) GPD EOF∆ ≅ ∆ 3) HL

56) Statements Reasons

1) Quad. DEFG is a parallelogram,

GP DF⊥ , EO DF⊥

1) Given

2) GPF∠ and EOD∠ are right angles 2) def of perpendicular lines 3) GPF EOD∠ ≅ ∠ 3) all right angles are congruent

4) 𝐺𝐺𝐺𝐺 ∥ 𝐷𝐷𝐸𝐸 4) def of parallelogram

5) 1 2∠ ≅ ∠ 5) if 2 ∥ lines are cut by trans, then alt. int. ∠ s ≅

6) GF DE≅ 6) opposite sides of a parallelogram are ≅

7) GPF EOD∆ ≅ ∆ 7) AAS

57) Statements Reasons 1) 𝑗𝑗 ∥ 𝑘𝑘, 4 10∠ ≅ ∠ 1) Given 2) 10 2∠ ≅ ∠ 2) if 2 ∥ lines are cut by trans, then corrsp. ∠ s ≅ 3) 4 2∠ ≅ ∠ 3) transitive property 4) 𝑙𝑙 ∥ 𝑚𝑚 4) if 2 lines cut by trans & corr. ∠ s ≅ , then lines ∥


1) M is the midpoint of GH , G H∠ ≅ ∠ , 1 3∠ ≅ ∠

1) Given

2) GH MH≅ 2) def. of midpoint

3) GKM HJM∆ ≅ ∆ 3) AAS

4) HK HJ≅ 4) CPCTC

5) 2 4∠ ≅ ∠ 5) if 2 sides of ∆ are ≅ , then ∠ s opp. those sides ≅


1) AC AD≅ , BC BD≅ 1) Given

2) AB AB≅ 2) reflexive

3) CAB DAB∆ ≅ ∆ 3) SSS 4) CAB DAB∠ ≅ ∠ 4) CPCTC

5) AM AM≅ 5) reflexive

6) CAM DAM∆ ≅ ∆ 6) SAS

7) CM MD≅ 7) CPCTC

8) CMD∆ is isosceles 8) def of isosceles triangle 60) Statements Reasons

1) 1 2∠ ≅ ∠ , 3 4∠ ≅ ∠ 1) Given

2) AX BX≅ 2) if 2 ∠ s of ∆ are ≅ , then sides opp. those ∠ s ≅

3) AXD BXC∠ ≅ ∠ 3) vertical angles are congruent 4) AXD BXC∆ ≅ ∆ 4) ASA

5) DX XC≅ 5) CPCTC

6) DXC∆ is isosceles 6) def of isosceles triangle 61) Statements Reasons

1) PU SR≅ , 1 2∠ ≅ ∠ , RQ UT≅ 1) Given

2) RU RU= 2) reflexive 3) PU SR= 3) def of ≅ 4) PU RU RU SR+ = + 4) addition prop of = 5) PU UR PR+ = , RU RS US+ = 5) segment addition postulate

6) PR US= 6) substitution

7) PR US= 7) def of ≅

8) POR STU∆ ≅ ∆ 8) SAS 9) Q T∠ ≅ ∠ 9) CPCTC


1) D & E are midpoints, BAC BCA∠ ≅ ∠ 1) Given

2) AC AC≅ 2) reflexive

3) AD DB≅ , BE EC≅ 3) def of midpoint

4) AD DB= , BE EC= 4) def of ≅

5) AD DB AB+ = , BE EC BC+ = 5) segment addition postulate

6) AB BC= 6) if 2 ∠ s of ∆ are ≅ , then sides opp. those ∠ s ≅

7) AB BC= 7) def of ≅ 8) AD DB BE EC+ = + 8) substitution 9) AD AD EC EC+ = + 9) substitution 10) 2 2AD EC= 10) distributive prop. 11) AD EC= 11) division prop of =

12) AD EC= 12) def of ≅

13) ADC CEA∆ ≅ ∆ 13) SAS PART IV: Algebraic Problems 63) 6 64) 68 65) 60 66) 𝑚𝑚∠1 = 40°, 𝑚𝑚∠2 = 50° 67) 70 68) 90 or 124 69) 158 70) 28 71) 156 72) 120 73) 20 74) 72 75) 110 76) 8x = , 6y = 77) 12 78) 44

79) 53

80) 120 81) 30 82) 150 83) 15 84) 60 85) 105

GEOMETRY SEMESTER TWO – REVIEW SHEET 1) The length of an 80∘ arc in a circle of radius 27 is

(A) 162π (B) 81π (C) 12π (D) 6π (E) None of these 2) A circle circumscribes an equilateral triangle. If the diameter of the circle is 28, then the altitude

of the triangle is (A) 14 (B) 14 3 (C) 21 (D) 24 (E) None of these

3) The diagonal of a cube has length 18. The total surface area of the cube is

(A) 648 (B) 5832 (C) 648 3 (D) 216 (E) None of these 4) A rhombus has diagonals of lengths 12 and 16. Find the length of the altitude of the rhombus.

(A) 2.4 (B) 245

(C) 9.6 (D) 10 (E) None of these

5) Given: EA AC⊥ and BD CE⊥ . Find ED .

(A) 15 (B) 5 (C) 7 (D) 12 (E) None of these

6) Trapezoid ABDE is inscribed in circle C . Which of the following can NOT be true? (A) AB DE= (B) AD contains C (C) A E∠ ≅ ∠ (D) AD BE= (E) None of these

7) Find the area of an isosceles trapezoid if the shorter base has length 10, the legs have length 8, and one angle has measure 60°. (A) 56 3 (B) 44 (C) 72 3 (D) 112 3 (E) None of these

8) The ratio of the width to the length of a rectangle is 5:8. If the perimeter of the rectangle is 78

cm., then the area of the rectangle is (A) 1440 (B) 720 (C) 360 (D) 40 (E) None of these

9) Find m ABE∠ of ⊙𝑂𝑂 if 𝑚𝑚∠𝐶𝐶 = 24° and 𝑚𝑚𝐵𝐵𝐵𝐵� = 36° .

(A) 60 (B) 24 (C) 48 (D) 42 (E) None of these

A

B

C

DE

O

9 6 8

E

D

CBA

10) Find the area of a circle if its circumference is 16π . (A) 256π (B) 16π (C) 8π (D) 64π (E) None of these

11) Quad ABEF is a parallelogram. If 10AF = and 6FE = ,

and 7DE = , find BC .

(A) 187

(B) 95

(C) 215

(D) 94

(E) None of these

12) In ABC∆ , 𝐵𝐵𝐵𝐵��⃗ bisects ABC∠ , 12AB = , 15BC = ,

and 18AC = , find CD . (A) 6 (B) 8 (C) 10 (D) 14 (E) None of these

13) Two concentric circles have center C . If 5CA = , 3AB = ,

𝑚𝑚𝐵𝐵𝐵𝐵� = 45°. Find the area of the shaded region.

(A) 39π (B) 25 38

4π − (C)

398π

(D) 34π (E) None of these

14) The altitude to the hypotenuse of a 30∘ − 60∘ − 90∘ triangle has length 12. The longer leg of the

triangle has length (A) 24 (B) 12 3 (C) 6 3 (D) 4 3 (E) None of these

15) Find the length of the apothem of a regular hexagon if the area of its circumscribed circle is 64π .

(A) 16 2 (B) 8 3 (C) 8 (D) 4 2 (E) None of these

16) The ratio of the surface area of two similar cones is 2:3. Find the ratio of their volumes.

(A) 4:9 (B) 2:3 (C) 2 6 : 3 (D) 2 6 : 9 (E) None of these

E

D

F

CBA

A D C

B

17) Find the length of DE in circle O if 5AB = , 4BC = , and 12EC = .

(A) 53

(B) 6 (C) 3

(D) 9 (E) None of these 18) Find the length of CE in circle O if AC DE⊥ ,

12AC = and 3DB = . (A) 3 10 (B) 3 13 (C) 6 5

(D) 6 2 (E) None of these

19) AC is perpendicular to both DE and FG . If the perimeter of 18ADE∆ = , 4AB = , and 3BC = , then the perimeter of AFG∆ is

(A) 272

(B) 24 (C) 727

(D) 632

(E) None of these

Multiple Choice Problems in Textbook

20) Chapter 5: Page 630 #1-12 all 21) Chapter 7: Page 632 #1-12 all 22) Chapter 8: Page 633 #1-6 all, 11, 12 23) Chapter 9: Page 634 #1-14 all 24) Chapter 11: Page 636 #1-7 all, 10, 11, 13-16 all 25) Chapter 12: Page 637 #1-14 all

A

B

C

DE

O

G

D

F

E

C

B

A

O

D

C

E

BA

26) State whether a given quadrilateral ABCD is a rectangle, rhombus, square, trapezoid or an isosceles trapezoid. (A) 𝐴𝐴𝐵𝐵 ≅ 𝐵𝐵𝐶𝐶,𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐶𝐶 ≅ 𝐵𝐵𝐵𝐵

(B) 𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐵𝐵 ≅ 𝐵𝐵𝐶𝐶

(C) 𝐴𝐴𝐵𝐵 ≅ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐵𝐵 ≅ 𝐴𝐴𝐵𝐵, 𝐴𝐴𝐵𝐵 ≅ 𝐵𝐵𝐶𝐶

(D) 𝐴𝐴𝐶𝐶 ⊥ 𝐵𝐵𝐵𝐵, 𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐶𝐶 ≅ 𝐵𝐵𝐵𝐵

(E) ∠𝐵𝐵 ≅ ∠𝐶𝐶, 𝑚𝑚∠𝐶𝐶 = 90°, 𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶

(F) 𝐴𝐴𝐵𝐵 ∥ 𝐵𝐵𝐶𝐶, 𝐴𝐴𝐶𝐶 ≅ 𝐵𝐵𝐵𝐵

27) Given: Quad ABCD is a parallelogram (A) If 15AX = , then AC = ?

(B) If AD DC≅ , then m AXD∠ = ?

(C) If AD AB≅ and 𝑚𝑚∠𝐵𝐵𝐴𝐴𝐵𝐵 = 70°, then m DCA∠ = ?

(D) If 5 10AD x= + and 8 1BC x= + , then x = ?

(E) If 2 50m DAC x∠ = − and 10m ACB x∠ = − , then x = ?

28) If 12XY = and 15RS = , then MN = ? 29) If 3 1AC x= + and 3RS x= + , then RS = ?

CD

B

X

A

S

Y

R

X

MN

R

A C

S

B

30) Given: 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ∼ 𝛥𝛥𝛥𝛥𝑅𝑅𝑅𝑅

(A) If 3RM = , 4MS = , 5NT = , and RT = ?

(B) If 3RM = , 4MN = , 4MS = , and ST = ?

(C) If 𝑚𝑚∠𝛥𝛥𝛥𝛥𝛥𝛥 = 60° and 𝑚𝑚∠𝛥𝛥 = 40°, and 1m∠ = ?

(D) If the perimeter of 12RMN∆ = 6ST = , 8RS = , and 4RT = , then RM = ? and RN = ?

31) State whether the triangles are similar. If similar, state by what postulate of theorem they are similar.

(A) (B) (C)

(D) 32) Given: In ABC∆ , 𝐵𝐵𝐵𝐵��⃗ bisects ABC∠ .

(A) If 8AB = , 3AD = , and 12BC = , then DC = ?

(B) If 6AB = , 16AC = , and 18BC = , then DC = ?

33) Given: Quad ABCD is a parallelogram.

Find the value of x and y . 34) Find the value of x and/or y in each of the following. (A) (B) (C) (D)

6

x4

45°

10

x

y

30°

5

x

y

30°

9x

y

1

M

S T

N

R

ED

C

BA

15

20

40

30

20

16 2430

36DC

BA

15

925

D

CBA

60

704070

EF

D

B

CA

A DC

B

6

8

18

y

x

BA

CD

35) (A) If 8DE = and 2DM = , then DF = ?

(B) If 3DM = and 8ME = , then DF = ? (C) If 6DF = and 4DM = , then ME = ?, FM = ?, and FE = ? (D) If 2FM x= , 1DM x= + , and 4 3ME x= − , then x = ?

36) State whether a triangle having the given side lengths is acute, right, or obtuse. (A) 4,6,8 (B) 8,10,12 (C) 6,2 6,3 2 37) Find the lengths of the diagonals of a rhombus if its perimeter is 64 and one angle has measure 120°. 38) Given: 𝐴𝐴𝐵𝐵 is the diameter of circle P and 𝑋𝑋𝑋𝑋�⃖��⃗ is tangent at X . 𝑚𝑚𝐵𝐵𝑋𝑋� = 120∘

(A) 1 ?m∠ =

(B) 2 ?m∠ =

(C) 3 ?m∠ =

(D) 4 ?m∠ =

(E) ?m AXB∠ =

39) Given: 𝐸𝐸𝐸𝐸 is tangent to circle P at F.

(A) If 𝑚𝑚𝐴𝐴𝐶𝐶� = 100∘and 𝑚𝑚𝐵𝐵𝐵𝐵� = 40∘, then 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐶𝐶 =?

(B) If 𝑚𝑚𝐸𝐸𝐵𝐵� = 60∘and 𝑚𝑚𝐵𝐵𝐶𝐶� = 100∘, then 𝑚𝑚∠𝐵𝐵𝐸𝐸𝐸𝐸 =?

F

EM

D

YX

BP

A

34

21

D

E

B

P

A

K

C

F

40) Given: Circle O with tangent 𝐴𝐴𝐵𝐵��⃗ , 𝑂𝑂𝐸𝐸 ⊥ 𝑋𝑋𝐶𝐶

(A) If 15OA = and 12AB = , then the circle has radius ?=

(B) If 13OF = and 12XC = , then ?OG =

(C) If 3GF = and 10XC = , then the circle has diameter ?=

(D) If 25OB = and 30XC = , then ?OG =

41) Given: Circle X with tangent 𝐵𝐵𝐴𝐴��⃗

(A) If 𝑚𝑚𝐵𝐵𝐸𝐸� = 25∘and 𝑚𝑚𝐶𝐶𝐸𝐸� = 45∘, then 1 ?m∠ =

(B) If 4, 3,CY YE= = and 2DY = , then ?YF =

(C) If 𝑚𝑚𝐶𝐶𝐸𝐸� = 85∘and 𝑚𝑚𝐵𝐵𝐸𝐸� = 37∘, then 2 ?m∠ =

(D) If 𝑚𝑚∠2 = 53∘and 𝑚𝑚𝐵𝐵𝐸𝐸� = 24∘, then ?m CDF∠ =

(E) If 13CB = , 5DB = and 6BE = , then ?EF =

42) Given: A regular hexagon inscribed in circle P . If 10AB = , find:

(A) the apothem of the hexagon

(B) the circumference of circle P

(C) the area of the hexagon

(D) the area of circle P

(E) the area of the segment bounded by 𝐴𝐴𝐵𝐵 and 𝐴𝐴𝐵𝐵�

B

X C

O A

G

F

3

12Y

B

F

C

X

A

D

E

P

BA

43) Find the area of:

(A) a square with diagonal of 5

(B) a circle with circumference 12π

(C) a right triangle with one leg of length 12 and hypotenuse of length 13

(D) an equilateral triangle with perimeter 24

(E) a parallelogram with 2 consecutive sides of length 8 and 10 which have an included angle of

measure 60°

(F) an isosceles trapezoid with legs of length 5 and bases of lengths 6 and 14

44) Find the lateral area, total area, and volume of each of the solids described:

(A) a right square pyramid with height 12 and base edge 10

(B) a right circular cylinder with radius 5 and height 12

(C) a right regular hexagonal prism with height 10 and base edge 12

(D) a right circular cone with radius 10 and height 24

(E) a sphere with a great circle of area of 16π

45) If a radius of a sphere is tripled in length, then its volume is multiplied by a factor of . 46) Two similar cones have volumes 12π and 96π . If the lateral area of the smaller cone is 15π , what

is the lateral area of the larger cone? 47) A square is circumscribed about a circle. What is the ratio of the area of the square to the area of

the circle? 48) A cube is inscribed in a sphere. Find the volume of the sphere if the cube has a total area of 24 cm.

Geometry Semester 2 – ANSWERS 1) C 2) C 3) A 4) C 5) C 6) B 7) A 8) C 9) D 10) D 11) A 12) C 13) C 14) A 15) E 16) D 17) D 18) C 19) D 20) Ch. 5, pg. 630

1) A 7) A 2) D 8) C 3) A 9) B 4) D 10) C 5) B 11) C 6) B 12) B

21) Ch. 7, pg. 632 1) C 7) B 2) D 8) A 3) A 9) B 4) D 10) D 5) B 11) C 6) B 12) B

22) Ch. 8, pg. 633

1) A 7) Skip 2) C 8) D 3) C 9) Skip 4) D 10) Skip 5) B 11) C 6) A 12) B

23) Ch. 9, pg. 634 1) C 8) B 2) A 9) B 3) C 10) D 4) C 11) A 5) C 12) D 6) C 13) A 7) A 14) D

24) Ch. 11, pg. 636

1) A 9) B 2) D 10) C 3) B 11) C 4) C 12) Skip 5) B 13) B 6) A 14) D 7) D 15) D 8) Skip 16) B

25) Ch. 12, pg. 637

1) D 8) B 2) B 9) A 3) A 10) D 4) C 11) C 5) B 12) A 6) B 13) D 7) A 14) B

26) (A) rectangle (B) isos. trap. (C) rhombus (D) square (E) trapezoid (F) isos. trap. 27) (A) 30 (B) 90 (C) 35 (D) 3 (E) 40 28) 9 29) 8

30) (A) 354

(B) 283

(C) 80

(D) 16 83 3

,

31) (A) NEI (B) ∆ABD~∆BCD by SSS similarity (C) ∆ABD~∆DBC by SAS similarity (D) not similar

32) (A) 92

(B) 12 33) x = 2, y = 6 34) (A) 2 5

(B) = =x y 5 2

(C) = =x 5 3 y 10,

(D) = =x 3 3 y 6 3, 35) (A) 4 (B) 33

(C) 5,2 5,3 5 (D) 3 36) (A) obtuse (B) acute (C) right 37) 16,16 3 38) (A) 120 (B) 60 (C) 60 (D) 60 (E) 90

39) (A) 70 (B) 70 40) (A) 9 (B) 133 (C) 34

3

(D) 20 41) (A) 35 (B) 6 (C) 24 (D) 65 (E) 29

6

42) (A) 5 3 (B) π20 (C) 150 3 (D) π100

(E) π −50

25 33

43) (A) 252

(B) π36 (C) 30 (D) 16 3

(E) 40 3 (F) 30 44) (A) 260, 360, 400 (B) π π π120 170 300, , (C) +720,720 432 3 2160 3, (D) π π π260 360 800, ,

(E) π π25664 ,

3

45) 27 46) π60 47) π4 : 48) π4 3

bxam1nar1u1 1a

Chapter 1 . . the appropriate letter·

Indicate the best answer by writmg . t are not coplanar?

1. Which of the following sets of pom s b. K, o, G, E

a. E,H,O,G d. H,K,0,J c. E, 0, F , J d .

. t re containe m more 2. Which of the following sets of pom s a

than one plane? G G

b. £ , 0, a. '0, J 0 H

H E G d. G, ,

c. ' ' ~ 3. How many planes contain point E and J K?

a. o b. exactly 1 d. unknown c. unlimited

4. If GH bisects EF, which statement is not necessarily true?

a. 0 is the midpoint of GH. b. EO ::::: OF c. E, F, G, H, and Oare coplanar. d. GO + OH = GH

5. Points A, B, C are collinear, but they do not necessarily lie on a line in

the order named. If AB = 5 and BC = 3, what is the length of AC? a. either 2 or 8 b. either 2 or 4 c. 2 d. 8

6. On a number line, point R has coordinate -5 and point S has coordinate 3. --+

Point X lies on SR and SX = 5.. Find the coordinate of X. a. - 10 b. - 2 c. 8 d. O

7. Which angle appears to be obtuse? a. LAEB b. LDEB c. L CEA d. LAED

--+ --+ 8. If EC bisects LDEB, EB bisects LDEA, and

m L BEC = 28, find the measure of L CEA. a. 28 b. 56 c. 84 d. 112

9. Which two angles are adjacent angles?

a. L DEB and L BEA b. L DEB and L CEA c. LDEC and LBEA d LDEA

· and LDEC

E D

Exs. 7-9

10. M is the midpoint of yz If YM _ a. 7 b. 10 ·

20- r + 3 and YZ = 3r - 1, find MZ.

. c. d. 4 11. Which of the following is not l

a. Exactly one plane contai a1. ways true when lines j and k intersect?

b Th 1. . ns me j • e mes mtersect in exactl o . .

c. All points on j and k Y ne pomt. d . are coplana .

. Given any point p . r pomts . · on J and any ·

pomts. pomt Q on k, P and Q are collinear

626 I Examinations

E

,..~

chapter 2

Indicate the best answer by writing th . e appropriate letter.

1. If m L 1 = 60 and m L 2 _ 30 th following? ' en L 1 and L 2 cannot be which of the

a. acute .6 c. vertical .6

2. Given: If q, then r. conditional?

b. adjacent ~ d. complementary ~

Which of the f 11 · . o owmg is the converse of the given

a. r implies q. b .f 1 'f • r I q. c. q on y 1 r. d 'f . • r 1 and only if q.

3. What are basic mathematical assumptions called?

a. theorems b. postulates c. conditionals d. conclusions

4. Which of the following cannot be used as a reason in a proof? a. a definition b. a postulate . c. yesterday's theorem d. tomorrow's theorem

5. LA and LB are supplements , m LA = 2x - 14 , and m L B = x + 8. Find the measure of LB. a. 62 b. 30 c. 40 d. 70

6. If L 1 and L 2 are complements, L 2 and L 3 are complements, and L 3 and L 4 are supplements, what are L 1 and L 4? a. supplements b. complements c. congruent angles d. can' t be determined

7. The statement "If mLA = m L B and m L D = m L A + m L C , then m L D = m LB + m LC'' is justified by what property? a. Transitive b. Substitution c. Symmetric d. Reflexive

8. If TQ l. QR, which angles? a. L 2 and L 3 c. L 5 and L 8

angles must be complementary

b. L 3 and L 4 d. L 3 and L 7

9. If mL 8 = x + a. 100 - x

80, what is the measure of L 9? b. 100 + x

c. x - 80 d. x - 180

10. If QT l. PS, which statement is not always ~rue? a. L 8 :::::::: L 9 b. L 2 - L

d. L 8 and L 9 are supp . .6 · c. L 8 is a rt. L .

11. If SQ bisects L RST, which statem~nt m7us~ beLtrRueS~ ST b 2m L - m

a. 2 • mL6 = mLR d. LRST := L RQT c. L 4:::::::: L 6 •

s Exs. 8-11

Chapter 3 Indicate the best answer by writing the appropriate letter·

1. If BE bisects L ABC, what is the measure of L AEB? a. 30 b. 35 c. 40 d. 45

2. If mLABE = 40 what is the measure of LDEB? a. 140 b'. 40 c. 75 d. 135

3. If AB II DC, what is the measure of L D? a. 70 b. 80 c. 90 d. 100

4. Which of the following would allow you to conclude that

AD II BC? b. L ABE ::::: L BEC a. L DEC ::::: L BCE

c. L BEC ::::: L BCE d. m LA + m L AEC = 180

5. What is the measure of each interior angle of a regular octagon? a. 150 b. 144 c. 140 d. 135

6. The plane containing Q, S, A, U appears to be parallel to the plane containing which points? a. Q, E, K, S b. E, K, C, R Q

A

u

c Exs. 1-4

c. R , E , Q, U d. U,R, C,A IR ;:----

7. Which of the following appear to be skew lines? - - - -a. QE and AC b. QU and KC - - - -c. AC and UR d. QS and AC

/

E Exs. 6-8

-8. EK does not appear to be parallel to the plane containing which points? a. U, A , C b. Q, U, A c. Q, U, R d. Q , S , C

9. The sum of the measures of the interior angles of a certain polygon is the same as the sum of the measures of its exterior angles. How many sides does the polygon have? a. four b. six c. eight d. ten

10. What is the next number in the sequence 1, 2, 4, 7 , 11, ... ? a. 17 b. 13 c. 16 d. 15

11. AC is a diagonal of regular pentagon ABC DE. What is the measure of LACD? a. 36 b. 54 c. 72 d. 108

12. A, B, C, and D are coplanar points. m LACD = 2x + 8. Find the value of x.

AB II CD, AB J_ Ac, and

a. 41 b. 49 c. 90 d. 180 13. What i.s .t~e principal basis for inductive reasoning?

a. defm1t10ns b. previously proved theorems c. postulates d. past observations

628 / Examinations

K

~ chapter 4

111 Exercises 1-8 write a method (SSS --" to prove the two triangles co ' SAS, ASA, AAS, or HL) that can be u~ ngruent.

J. 2. 3.

4. 5.~ 6.

p

7. Given: PO .l plane X; OT = OS

8. Given: PO ..l plane X; PT = PS

Exs. 7, 8

Indicate the best answer by writing the appropriate letter.

9. In 6.RXT, LR:::: LT, RT= 2x + 5, RX= 5x - 7, and TX = 2x + 8. What is the perimeter of 6 RXT? a. 5 b. 15 c. 18 d. 51

10. If 6.DEF :::: 6.PRS, which of these congruences must be true? - - -- --

a. DF:::: PS b.EF::::PR c. L E:::: L S d. L F:::: L R

11. In 6.ABC, AB = AC, m LA = 46, and BD is an altitude. What is the

measure of L CBD? a. 23 b. 44 c. 67 d. 134

12. An equiangular triangle cannot be which of the following? a. equilateral b. isosceles c. scalene d. acute

13. Point X is equidistant from vertices T and N of scalene 6TEN. Point X

must lie on which of the following? a. bisector of LE b. perpendicular bisector of TN

c. median to TN d. the altitude to TN

14. Given: AB II DC; AB ::: CD; L 1 ::: L 2 Dr-----...,;

To prove that DE :::: BF, what would you prove first?

a. 6.ADE :::: 6.CBF b. 6.ABF ::: 6.CDE c. 6.ABC :::: 6.CDA d. cannot be proved

A B

Examinations / 629


1. Both pairs of opposite sides of a quadrilateral are parallel· Which special

kind of quadrilateral must it be? a. parallelogram b. rectangle c. rhombus

2. The diagonals of a certain quadrilateral are congruent. not be used to describe the quadrilateral?

d. trapezoid

Which term could

a. isosceles trapezoid b. rectangle c. rhombus d. parallelogram with a 60° angle

3. M is the midpoint of hypotenuse TK of right !::,.TAK. AM = 13. What is

the length of TK? a. 26 b. 19! c. 13 d. none of these

4. In o WXYZ, WX = 10. What does ZW equal? a. 16 b. YZ c. WY d. none of these

5. A diagonal of a parallelogram bisects one of its angles. Which special kind of parallelogram must it be? a. rectangle b. rhombus c. square d. parallelogram with a 60° angle

6. The lengths of the bases of a trapezoid are 18 and 26. What is the length of the median? a. 8 b. 22 c. 44 d. 34

7. In quad. PQRS, PQ = SR, QR = PS, and m LP = m L Q. Which of the following is not necessarily true?

- - - -a. PR .l QS b. PR :::::: QS c. LP:::::: LR d. LR:::::: LS

8. In 6ABC, AB = 8, BC = 10, and AC = 12. Mis the midpoint of AB ,

and N is the midpoint of BC. What is the length of MN? a. 4 b. 5 c. 6 d. 9

9. If EFGH is a parallelogram, which of the following must be true? a. LE :::::: L F b. L F :::::: L H

c. FG II GH d. mLE + mLG = 180

10. Which information does not prove that quad. ABCD is a parallelogram?

a. AC and BD bisect each other. b. AD II BC; AD :::::: BC

c.ABllCD; AD::::::Bc d.LA::::::LC· LB::::::LD '

11. In the figure, RU :::::: US and L 1 :::::: L 2. R Which of the following canno!_!?e proved? a. L 3 - L 4 b. RV :::::: VT

- -c. US :::::: VT d. ST = 2 · UV

12. Which of the following must be true for any trapezoid? a. Any two consecutive angles are supplementary. b. At least one angle is obtuse. s c. The diagonals bisect each other. Ex.11 d. The median bisects each base.

630 / Examinations


· le are in the ratio 3: 3: 4, what is l. If the measures of the angles of a tnang 1 "

l f the triang e t the measure of the largest ang e 0 d 90 a. 40 b. 54 c.

72 ·

· t proportion? 2. If 6ABC ~ /::,JOT, which of these ts a correc AC BC

BC JT AB AC AB == OT d. - = -a. AC = OT b. JT = JO c. BC JT JT OT

3. If ~ = ~ what does ~ equal? b y' b

x a. -

a b. ~

x

y c. -

x d. ~

y

4. 6ABC ~ 6DEF, AB = 8, BC = 12, AC == 16, and DE = 12. What is

the perimeter of 6DEF? a. 36 b. 40 c. 48 d. 54

5. Which of the following pairs of polygons must be similar? a. two rectangles b. two regular hexagons c. two isosceles triangles d. two parallelograms with a 60° angle

6. Quad. GHJK ~ quad. RSTU, GH = JK = 10, HJ = KG = 14, and RS = TU = 16. What is the scale factor of quad. GHJK to quad. RSTU?

a. ~ b ~ c I d t6 7 • 8 • 8 • IO

7. Which of the following can you use to prove that the two triangles are similar?

8.

a. SAS Similarity Theorem c. SSS Similarity Theorem

Which statement is correct?

b. AA Similarity Postulate d. Def. of similar triangles

Exs. 7, 8

6 8 5 8 a. - = - b. ~ = ~ c. 6 · 10 = 8x d. - = -IO x 8 IO y

9. What is the value of u? a. 8 b. 10 c. 16 d. 25

10. What is the value of z?

a. 25 b. 28 28 d. 70 c. -3 3

11. In 6APC, the bisector of L p meets AC at B. p A _ AB = 12. What is the length of BC?

36 a -• 5 b. 12 c. 20

12. If 6RST ~ 6XYZ, what is the ratio of m LS t

IO

30, PC

d. 32

50, and

z Exs. 9, 10

a. m LR: m L Z b l . l o m L Y? • · c. RS:XY fon d. not enough informa 1

632 / Examinations

" ,,,,-- chapter 8

Indicate the best answer by writing the appropriate letter.

1. The shorter leg of a 30° -60° -90° triangle has length 7. Find the length of the hypotenuse.

a. 14 b. 70 c. 70 d. VT4 2. The altitude to the hypotenuse of a right triangle divides the hypotenuse

into segments 25 cm and 30 cm long. How long is the altitude? a. 15\13 cm b. 15\15 cm c. 5V30 cm d. sy'SS cm

3. The hypotenuse and one leg of a right triangle have lengths 61 and 11 · Find the length of the other leg.

a. 36 b. 50 c. 60 d. \13842 4. Each side of an equilateral triangle has length 12. Find the length of an

altitude.

a. 6 b. 12 c. 6y2 d. 6\/3

5. One side of a square has length s. Find the length of a diagonal.

a. 2Vs b. s0 c. ~0 d. sv3 2

6. What kind of triangle has sides of lengths 12, 13, and 18? a. an obtuse triangle b. a right triangle c. an acute triangle d. an impossibility

7. In 6RST, m LS - 90. What is the value of sin T?

ST b RS c. RS a. RT • ST RT

RT d. RS

8. What is the geometric mean_ between 2 and 24? d. 4\/3 a. 48 b. 16\13 c. 4\16

f . 0 3 9. One acute angle of a certain right triangle has measure n. I sm n = 5'

what is the value of tan n°? 3

4 b 4 c. a. 3 . 5 4

d t f nd the value of x? 10. Which equation could be use 0 1 x

_ __::__ b. sin 32° = 16 a. cos 58°

18.9 x

d. tan 46° = I0.4

5 d. 3

x c. cos 44° - 10.4

- -C BD ..l AC at point D, BC = 9, and AC 11. In rt. 6ABC, AB J.. B •

Find the ratio of AD to DC· 7 9 9 16 c. - d. 7

~ 32°44° 16 10.4

12.

a. - b. 9 9 16 . 1 with sides of lengths x, x + 7, and

12. For what value(s) of x is a tnang e

x + 8 a right triangle? x = - 7 or x = 5 d. - 7 < x < 5 b - 5 c. a. x = -7 · x -

Examinations / 633


In Exercises 1-3, PT is tangent to 0 Mat T. -1. If m L TMA = 80, what is the measure of TBA?

a. 100 b. 80 c. 280 d. 145

2. If m L M = 80, m L p = 50, what is the measure of

LMAP? a. 140 b. 150 c. 160

3. If PA = 9 and AB = 16, what does PT equal? 25 a. 12 b. - c. 15 2

d. 170

d. 20

4. Suppose PS were drawn tangent to OM at point S. If .--.....

m L SPT = 62, find mST. a. 62 b. 236 c. 118 d. 242

5. How many common tangents can be drawn to two circles that are externally tangent? a. one b. two c. three d. four

6. Points A, B, and C lie on a circle in the order named. mAB = 110 --and mBC = 120. What is the measure of LBAC? a. 130 b. 65 c. 60 d. 55

7. Refer to Exercise 6. If point D lies on AC, what is the sum of the measures of L ABC and L ADC? a. 180 b. 170 c. 160 d. 130

8. R and S are points on a circle. RS could be which of these? a. radius b. diameter c. secant d. tangent

9. If mBC = 120 and mAD = 50, what is the measure of LX? a. 25 b. 35 c. 60 d. 70 --10. If mBC = 120 and mAD = 50, what is the measure of L 1? a. 60 b. 85 c. 90 d. 95

11. If AY = j, YC = k , and YD = 7, what does BY equal? jk b 7} 7k k a. -7 • - c. - d

k j • 7j -In Exercises 12-14, XA is tangent to O O at X.

12. Whic~ these equals m L AXZ?

13.

a. mXYZ b. m L OXM 1 -- .--.... c. 2mXY d. lmXZ If the radius of 00 is 13 and XZ = 24, what is the distance from 0 to chord XZ? a. 5 b. 8 c. 11 d. v'407

14. If OM = 8 and MY = 9, what does XZ equal? a. 6\12 b. 2v'17 c. \/i45 d. 30

634 / Examinations

x Exs. 9-11


. ·meter is 44. What is the area? 1. One side of a rectangle 1s 14 and the pen d. 420

a. 112 b. 210 c. 224 . . ? . . d · circle with radms 8.

2. What is the area of a square mscnbe 10 a 4

r;; d. 128 a. 3 2 b. 64 c · 64 V 2

3. The area of a circle is 257T. What is its circumference? a. 57T b. 107T c. l 2.57T

d. 507T

4. What is the area of a trapezoid with bases 7 and 8 and height 67 d 168 a. 90 b. 336 c. 45 •

5. A parallelogram and a triangle have equal areas. The base .and hei.ght _of the parallelogram are 12 and 9. If the base of the triangle is 36, fmd its

height. a. 3 b. 6

6. What is the area of trapezoid ABCD?

c. 9

a. 96 b. 120 c. 144 d. 192

7. What is the ratio of the areas of 6.AOB and 6.COD? a. \13: 1 b. \13: 3 c. 3: 1 d. 9: 1

d. 12

D 6

18 B 8. What is the ratio of the areas of 6.AOB and 6.AOD? Exs. 6-8

a. \13: 1 b. 3: 1 c. 9: 1 d. cannot be determined

9. What is the area of a regular hexagon inscribed in a circle with radius 8? a. 16\13 b. 96\1'3 c. 128\1'3 d. 192\13

10. In the diagram, what is the length of AB? a. 6\/2 b. 67T c. 37T d. 367T

11. In the diagram, what is the area of the shaded region? a. 97T - 36 b. 127T - 36 c. 97T - 18 d. 127T - 18

12. If a point is chosen at random in the interior of O O , what is the probability that the point is inside 6.AOB?

2 1 3 a. - b. - c. - d

7T 4 27T • 27T

13. A rhombus has diagonals 6 and 8. What is its area? a. 12 b. 24 c. 36 d. 48 What is the area of a circle with diameter 12?

2 • a. 247T b. l 27T c. l 447T

What is the area of an equilateral triangle with perimeter 24 ?

a. 64\1'3 b. 32\1'3 c 320 . 3

14.

15. d. 367T

d. 16\13

16. What is the area of a triangle with sides 15 15, and 24

? a. 54 b. 108 c.' 180 d. 2 16

636 / Examinations

Exs. 10-12

~~ chapter 12

Indicate the best answer by writin g the appropriat I t

1. What is the volume of a re t e e ter. 08 c angular sol'd . h . a. I b. 216 1 Wit dimensions 12, 9 , and 6?

2. What is the total surface a f c. 432 d. 648 rea o the I'd . a. 234 b. 468 so 1 m Exercise 1?

d. 360 . ·1 c. 252 3, Two s1m1 ar cones have hei hts

volumes? g 5 and 20. What is the ratio of their

a. 1:64 b. l: 4 · h c. I : 16 d 4 · I 6

4. What 1s t e volume of a re 1 · · height 6? gu ar square pyramid with base edge 16 and

a. 128 b. 256 c. 512 5. What is the lateral area of the p . d . .

a 256 yram1 m Exercise 4? . b. 320 c. 576

6. A sphere has area 167T. What is its volume?

a. 831T b. 3231T 647T c. -3

d. 1536

d. 640

d 2567T • 3

1. A cone has radius 5 and height 12. A cylinder with radius 10 has the same volume as the cone. What is the cylinder's height? a. 1 b. 2 c. 3 d. 4

8. A cube is inscribed in a cylinder with radius 5. What is the volume of the cube? a. 15\!2 b. 250\12 c. 125 d. 100

9. A plane passes 2 cm from the center of a sphere with radius 4 cm. What is the area of the circle of intersection? a. l 27T cm2 b. l 67T cm2 c. I 87T cm2 d . 207T cm2

10. Find the total surface area of a cylinder with radius 4 and height 6. a. 167T b. 327T c. 487T d. 807T

11. Two similar pyramids have volumes 27 and 125 . If the smaller has lateral area 18, what is the lateral area of the larger? a. 30 b. 83! c. 50 d. 25

12. The base of a right prism is a regular hexagon with side 4. The height of the prism is 6. What is the volume of the prism? a. 1440 b. 720 c. 48\13 d. 36\13

13. What is the lateral area of the prism in Exercise 12? a. 24 b. 36 c. 72 d. 144

14. Find the total surface area of a cone with radius 9 and slant height 12. a. 1087T b. 1897T c. 8 hr\17 d. 2 l 67T

Examinations / 637

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Geometry – Semester One Topics