book1-7

http://ibrahimkhalilmath.webs.com/ T.ibrahim khalil

1

For a good STarT & PracTice in

grade:7

Prepared By

T. ibrahim khalil


2

Powers (Prerequisites)

1) Complete this table:

Power

Base Exponent Developed form Value

42 4 2 4x4 16 3x3x3x3 7 1 5x5x5 2 5 1000 2x2x2x2 1 3 6x6x6

2) Complete:

102 = 100 3 .... = 9

10 .... = 1 000 7 .... = 49

10 .... = 10 2 .... = 64

10 .... = 100 000 000 5 .... = 625

10 .... = 1 2 ....3 .... =36

3) Write in expanded form: Example: 26.73 = 26 + 7 3+10 100

a) 8.321 = ....................................... b) 12.109 = ....................................... c) 1209.0906 = ...................................

4) Simplify using order of operations. Show all work!

A= 214 ÷ 7 5 -3 =----------------------------------

B=15 - ( 7 + 3 ) + ( 11 - 9 ) =----------------------------------

C= 8 2 - (3 + 9) + 8 ÷ 2 3 =----------------------------------


3

Powers 1) Find the value of each expression. Show all work.

A= 1 200200 -1 ; B=(32 – 22)5 + 2(3 + 15) ; C=32 + 2 (8 – 4)2 – 5(27 – 52) D=225-3 (1)5 - (32 - 23)1000 ; E= 2002 4 2001(72 - 8 9) 3 - 3 ÷ (7 - 3 2)

F= 2 2

28 -8÷2

6-2 ; 3 4 2 2015G = (4 -62) 5-16÷ (2) -(100-3 33)

2) Find x in each of the following cases:

x 5 8 3 3 3 5 x 5 xx

13 42000 7 x 8 x 2 8 6

x x

5 a) 7 × 7 = 7 b) =1 c) 5 × 4 = x d) 12 = 3 × 4 e) 12345 =155 100 f) (2×3-x) = 0 g) = 5 h) 4 =2 i) (r ) = r j) =105 10

3) Write as an:

A=

55

64

2

2 ; B = 253575; 4 9

5 310 10C =(10 )

; 2 3 D =16 64 ; 5 2 4

23 ×9 ×27E=

81

4) Simplify:

A=3a²b32ab² ; B=3(a²b3)24(a3b)3 ; C= 63

46

yx5yx20

;

52 3 4

23 4

a b aD = .

a b

5) Copy & complete:

a)

24 3

22 5

2 72 7

2 7 b)

8 4

2 220 4 2 58 25

6) Show that:

a) 4

2 3 5 -11+5+5 +5 =5-1

b) 17 17 17 176 4 3 8 (without calculating the powers). 7) Think, then calculate quickly: 3 3A 2 7 5 ; 3 5B 5 2 25 ; C = 7 72 2000 0.5


4

Prime Numbers (Prerequisites)

1) Perform the operation then find the divisor, quotient & the remainder in each case: a) b) c) 702 ÷ 4

2) a. List the factors (divisors) of 10. …………………………………… b. List the factors of 15. …………………………………… c. What is the greatest common factor (GCF) of 10 and 5? …………………………………… 3) a. List the first 8 multiples of 2. ……………………………………………………… b. List the first four multiples of 7. ……………………………………………………… c. What is the lowest common multiple (LCM) of 2 and 7? …………………………………

4) Write "T" or "F":

5) Write the divisors of:

3:---------- 7:---------- 13:---------- What do you conclude? What do you call such numbers?

---------------------------------------------------------

Divisible by 2 3 5

72 3660 305

5 9 6 8

4 2 5 1 1

5

1 -

-

- 3


5

Prime Numbers

1) Given the numbers a=100 & b=160. a) Write the prime factorization of a & b. b) Determine the GCD(a;b) & LCM(a;b).

c) Compare the two numbers:

ab & LCM(a;b)GCD(a;b).

d) Simplify, write the result in exponential form:

A=ab ; B= a3b2 ; C=5

5ba

2) a) Write the prime factorization of 22750 & 3850. b) Deduce the GCD & LCM of 22750 & 3850. c) What is the prime factoring of 22750100 ? 3) The prime factorization of 200 can be written as: 200 = 2x5y .

Calculate x + y . 4) Fill in the blanks so that the number is divisible by 3 and 5 at the same time: (Give the all possible solutions). 6__341__ 5) Find the GCF & LCM of: a) 10x3y and 15x2y2 b) a2b and ab2

6) Use the Euclidean division to calculate the GCD of:

7) Write down two positive numbers with GCD=2 and LCM=30.

806 & 496 a b r a= bq+r


6

A

B

C

D

E

F G 70°

O

T

A

P C S

N

Geometry Review 1) Refer to the figure at the right.

a) Name:

an acute angle:-------

an obtuse angle:------

a right angle:-------

a straight angle:----------

a pair of adjacent complementary angles:-------------

a pair of adjacent supplementary angles:--------------

b) How are the lines (AD) & (FC)?

c) Calculate:

BGC=------------ ; AGF=----------- ; FGE=-------------

2) For each diagram find the value of the unknown. a) b) c)

3) Given: [SQ) bisects RST. [SR) bisects PST and m1 = 17°. Find: PSR and RST .

4) Determine the nature of each triangle then calculate its angles: a) triangle CPA b) triangle TPO c) triangle OSN

x 40y

75x

O

A B

C

a

a

a

P R

T

Q

S

1 2


7

118C 40

BA

E

D

F

yx

81

C

FB

A

D Ey

x

40

28

5) Fill in each blank with the correct word: angle bisector, median, perpendicular bisector, or altitude(height).

a) It is a line that runs from a vertex of a triangle to the midpoint of the opposite side. ___________ b) It cuts an angle of a triangle into two congruent angles. ________________ c) It is a line that goes from a vertex of a triangle and is perpendicular to the opposite side. _________ d) It is a line that is makes angle 90º at the midpoint of each side of

a triangle. _______________ 6) a) Draw: ABC where AC=6 cm, A = 40° & B = 60° . [BH] is a height.

Calculate ABC & ABH . b) Draw: GHI isosceles of vertex H where GH=5 cm & H =80° . Calculate the other angles of GHI. c) Draw: MNO right at O where MN=4.5 cm & N = 40° . [OI] is a median. Calculate: OMN & IM. d) Draw: VWZ equilateral where VW=3 cm. [VT] is the bisector of V .

Calculate ZWV & ZVT .

7) For each figure, find the value of the unknowns.

a) b) c)

8) On the opposite figure we have: [FE) is the bisector of AFM . ˆEFM = 20° & ˆFAM =50°. a) Calculate the measure of AFE & AFM . b) What is the nature of triangle AMF?

Justify the answer. 9) Given: M is the midpoint of [AN]; N is the midpoint of [MB]. Prove: AM=NB.

x

M

P

N

41°

A

M

FE

BM NAX X


8

21A

D

C

B

54 cm

11 cm

12 cm 12 cm

24 cm

11 cm A B

C D

E

F

H

G

A DCB

E

x y

10) Given: 3 4 Prove: MA = MC 11) Given: C is the midpoint of [BD] .

1 2 . Prove: AB=CD.

12) In the figure, EB EC. Prove that x y. 13) ABCD is a rectangle. a) Calculate the area & the perimeter of ABCD. b) Calculate the area of triangles GDH & FBE. c) Deduce the area of the colored surface. 14) a) What are the coordinates of the points A, B, C & D? b) Calculate the area of the:

i. rectangle ABCD. ii. square DMON. iii. right triangle AFE.

c) Calculate the area of the interior uncolored part of the rectangle ABCD.

M

CA1 23 4

A B

C D

E M O

N

1 x

y F


9

C

A

B

D

F

E

Q

P

R

40

454045

40

95

A

C

D

B E

Congruent Triangles

1) Which pair of triangles must be congruent? Justify the answer.

a)

b)

2) State the additional piece of information needed to prove congruence of the triangles by the method stated.

a) b) c)

3) Given: A =D , B is the midpoint of [CE]. a) Prove: ABC DBE. b) Write the homologous elements. c) Show that BAE =BDC .

C

BA

D

F

EP

Q

R

E

F G H

SAS

I

J

K

L

ASA

Q

S

T

U R

SAS


10

A

E

B C

D

A

DC

B EF H

4) Given: AB=AC & EB=DC. a) Prove: BEC CDB. b) Show that ACE =ABD c) M is the midpoint of [BC]. Prove the EMD is an isosceles triangle. 5) Consider the triangle ABC where 0 0B = 40 , C =50 and BC =5cm . [AH] is the altitude relative to [BC]. a) Draw the figure. b) Calculate the measure of A , then deduce the kind of ABC c) Find the measure of the two angles ˆ ˆBAH and CAH . d) Are the two triangles AHB and AHC congruent? Why? 6) a) What are the indicated properties on the figure?

b) Prove that ABC AED. c) Prove that ACD is an isosceles triangle. d) i) Show that BCD =EDC ii) Deduce that BCD EDC.

e) Prove that CF=DH. f) Prove that AFH is an isosceles triangle.

7) Given: NGI NAI. [IN] bisects GIA .

a) What is the nature of AIG? Justify the answer.

b) Prove that N is the midpoint of [AG].

c) Prove: GT=AT.

d) Show that [NI] is the bisector of GTA e) Prove that (IN) (AG).

8) Given: [OP] bisects RON.

NO = OR. a) Prove that ONT ORT. b) Deduce that 5 = 6. c) Show that NPR is an isosceles . d) E & F are respectively the midpoints of [ON] & [OR]. Prove that [TO) is the bisector of ETF .

G

I T

A

N 1 2

3

4

TP O

N

R

6 5

4

3 21


11

Signed Numbers (Prerequisites)

1) Consider the axis of origin O & unit 1cm.

a) What is the abscissa of each point? A( ..... ); B( ..... ); C( ..... ); D( ..... ); E( ..... ); F( ..... ) b) What relation exists between the abscissas of R & E?

c) What is the midpoint of [RE]? What is the midpoint of [DE]?

d) What is the abscissa of the midpoint of [RI]?

e) Calculate the distances:

AE = ------- ; DA = ----------

f) Compare: 2--- 5 ; -3 --- 0 ; -7 --- -5

2) Write the sum that the following moves on a number line represent then

find the result.

(---) + (---) + (---) = ----

3) Calculate:

A= 32÷4 - 2 + 73 =----------------------------------------- B= (3 + 57)÷ 2 +1=------------------------------------------ C=100 - [7x( 5 - (8 - 3))] =-----------------------------------

0 5 start end

D

-3

I

O +5

A E S R


12

Signed Numbers 1) Calculate each of the following expressions, show all steps of calculations:

a= 6 – (-3) + (-5) -7 ; b=-7 – 8 + (-9) - (-10) ; c=–36 ÷ 12 ÷ 3; d= 3 (-6) – 4 (-3) + (-5) (2) ; e= 4 5 3 2 ; f=

2 2 2 23 3 5 10 ( 2) ;

g= –2 (–14) ÷ (–7) ; h= (–1)5 – (–2)4 + (–3)3 + (+4)2 – (–5) ;

(6-3) (-9 +5)i =(7 -9 +1) 2

;

6-4 5+8j=3+7 (-2) +7

; k=-(-2+8–1) – (7+6–8) – (-3+1) ; l=(-13)-3004 -2(-1502).

2) Find the missing number.

a) ___ + 8 = – 12 b) -6__ =-12 c) –16 ÷ ___ =-(–2) d) ___ – (–8) = 4

3) Insert <, >, or = symbols to make a true statement.

a) -18 ÷ -3 __ -24 ÷ -4 b) -5(-4)2 __ -4(-5)2 c) 5(18 – 24) __ 90 - (-120) d) (-10)2-(10) 2 __ (-1)4 +(-1)5

4) a) Evaluate each expression for a -2 & b +3:

A 2b+3a-2 ; B(2b+3)(a- 2) ; C (a - 2b) (-1 + a)

a + b + 1 ; D b2 - 3a2 + 10

b) List A, B, C & D in descending order. 5) Four youngsters are in the subbasement of an apartment building. They decide to play in the elevator. They go up 3 floors, then up again 4 floors. They go down 6 floors, up 8 floors, and down two floors before the janitor catches them and ends their little game. a) Write a numerical expression that represents the problem. b) Where were they when the janitor caught them? 6) Given: x= -3 + 10÷(-5) ; y= (-2)2-[-(-4)] ; z= -1 - 43 – (-10) a) Calculate x, y & z. b) List x, y & z in ascending order.

c) Evaluate: A=x+y+z ; B=3x -10z + y100 ; x +zC =x-z


13

118C 40

BA

E

D

F

yx

A B

CD

Angles & Lines (Prerequisites)

1) Find the unknowns in the following figure.

x = -------- y = --------

2) Use the figure to complete the statements with:

Parallel, Confounded, Secant (not perpendicular) or Perpendicular.

3) What are the cases of congruent triangles? ----------------------------------- 4) Given: AC=DB. a) Prove: ABC DCB .

--------------------- --------------------- --------------------- ---------------------

b) Name two pairs of // lines. ------------ ------------

(AB) & (D) are ………………………………………… (D) & (d) are ……………………………………………… (d’) & (xy) are ……………………………………………… (D) & (AC) are ……………………………………………… (d’) & (AB) are ………………………………………………… (D) & (D’) are ……………………………………………………

B

E

A

C

(d')

(d)

(D)

x y

(D')


14

17

14 15 16

13 12 11

10 9

8 7

6 5 4 3

2 1

(p)

(q)

A

B

D

M C

55 105

75

55

100

A B

D

FE

C

2x

40

x

1 10

E

C

A

F

G

B

D

F

D

C

A

B 1

E

Angles & Lines 1) Given: (p)|| (q) ; 1 = 107 ; 11 = 48. Find the measure of all indicated angles. 2) In the figure at right, find the numbered angles. 3) a) Prove that (AD)//(BC). b) Calculate: ABD ; BDC; & ADC . c) Deduce that MA=MD.

4) Find the unknowns in the following figures. a) b)

5) a) What are the indicated codes on the figure? b) Prove: ΔAWX ΔBYZ .

c) Deduce that (AX)//(BZ). d) Prove that (AB)//(WZ). 6) Given: [BE] bisects [CF] and (AE)//(BC). a) Prove that [CF] bisects [BE]. b) Show that (EC)//(BF).

1 2

32° 35° 3

4

W X Y Z

A B


15

Fractions (Prerequisites)

1) Write in simplest form:

46 =----- ;

2142 =------------ ;

420168 =---------

2) Complete the equivalent fractions:

2 4 ... 40 ... ...= = = = = .9 ... 27 ... 1800 81

3) Calculate & reduce:

a) 3 2+7 7

=--------------------- b) 5 23 3

=----------------------.

4) Complete the steps of calculations:

5 74A = +10 100

5 ...... 74= +10 ...... 100

...... 74= +100 100

......=100

5 3 5 ...... 3 ...... ...... ...... ......B = - = - = - =6 10 6 ...... 10 ...... ...... ...... ......

C= 8 7+25 10

=------------------------------------------------

D= 2 1-3 4

=-------------------------------------------------

E= 5 912 10

=----------------

F= 5 1÷2 4

=---------------------

5) Complete the table:

Fraction Decimal

12

0.25


16

Fractions 1) Write as irreducible fraction:

a)

3 3

32 5 7 112 5 7 11

b)

44 42 156196 260 99

c) 1232364

2) a) Calculate, write the answer in simplest form:

1 1 1A 12 3 4

; B = 34

45 +

75 ;

2 5C 13 3

;

1 3D5 10

E = 85

34 - 2

310 ;

1 3 1F = + ÷ 1 +4 4 8

; G =

25 2-6 3

; H =

25 2-6 3

b) Among the above fractions, which is a decimal fraction?

3) 9009 2 3Given : A=10395 5 2

.

a) i) Write the prime factorization of 9009 & 10395.

ii) Deduce the simplest form of 900910395

.

b) Write A as irreducible fraction. 4) Complete the table:

Decimal form Fraction form Fractional decomposition

17.52

0.000004

25

1 000

7.789

600 + 2 + 3

1000

5) Given the numbers: 93 216 625 ; ; ;8 75 111

a) Decompose each fraction into integral part + decimal part. b) Indicate the decimal & non decimal fraction.

c) Give the approximation to nearest 0.1 & 0.01 for the non decimal fractions.


17

O x

A

B

C

D

E

y

Graph (Prerequisites)

1) Consider the figure:

a) Find the abscissa of: A( ) ; D( ) ; O( ) ; I( ); S( ) & E( ). b) Find the abscissa of R the midpoint of [AD]. R( ). c) Calculate: OD=--------- ; DE=------------- d) Draw M & N the orthogonal projections of B & C on the (x'x) respectively. Find the abscissa of M( ) & N( ).

2) a) Write the coordinates of the drawn points:

b) Draw the height [CH] in ABC. What are the coordinates of H? c) Calculate the area of ABC. Area=------------

x

x

x

D

-3

I

O +5

A E S

B

C

x'


18

Graph

1) a) Plot A(1;1), B(1;4), C(2;1) and D(2;4) on a rectangular coordinate plane.

b) Join the points in the given order. Which letter is formed?

c) (AD) cuts (BC) in I. What are the coordinates of I? d) Prove that ABI CDI. e) What is the nature of ABDC? Calculate its area & perimeter. 2) a) Plot A(0;-4) , B(4;0) & C(4;-4) in an orthonormal system of axes x'Ox , y'Oy. b) What is the nature of OACB? Calculate the area of OACB. c) Let I be the midpoint of [AB]. Calculate the coordinates of the point I.

3) a) Plot P(3,3), Q(3,3) & R(3,3) on orthonormal system of axes x'Ox,y'Oy

b) Calculate the area of PQR.

c) S is the orthogonal projection of P on (x'x).T is the orthogonal projection of Q on (y'y). i) What are the coordinates of S & T? ii) Prove that OSP OTQ. Deduce the nature of POQ. d) Plot the point E so that PQRE is a rectangle. What are the coordinates of E? Calculate the area & perimeter of PQRE. 4) CDEF is a rectangle.

a) Write down the coordinates of D, E and F. b) Find the distances CD, BE, & DA. c) Find the area of: ∆CBF, ∆ABE, ∆CDA & ∆ABC.

A(-6,2)

B(4,-5)

C(7,7)

E F

D

y

x


19

H

59

E

B

A

D

G

I

yx

w

z

C

F

Midyear Review

1) Find the value of the expression. Show all work.

2 2000A = 4.25 4 -(6-4 1.25) ; B=225-3 (-1)5 - (32 - 23)10 ; C= 2

16-2 310 (4 -15)

D 5 - [4 - (1 - 9)] ; -9+6-5E =3-(6-8)

; F = [-1 – (4 – (- 2))] + (-2 + 3)-100

G = – 7 – 3×{5 – [8 + 12 (–2)]} – [–6 – (–4)×(–3)] + 36 (–2)

2) Simplify:

2 3 4 6

2 3 4 72 5 (2 ) 5A

(5 ) 2 2

; B=yx2yx14

2

46

; C=4 2 2

3 4(3a b )3a b

; D= 5 2

4 810 1010 10

3) Complete:

5 5.... .... ....

3 2 214 3 =2 3 7

16 49 9

4) Determine x in each case:

a) 3

7x

5 55

b) x 84 2 c) 3 3 2 54 7 28 x d) x + 7 = – 13

5) Given the numbers: A 2520 & B 4116 .

a) Write A & B as product of prime factors.

b) Calculate the LCM & GCD of A & B. Write AB

as irreducible fraction.

c) Calculate: 1 1C2520 4116

.

6) a) Calculate A, B & C. Write the results as irreducible fractions.

A= 7 5 13 6 ; B=

34 -

56

32 ; C = (

19 -

35 ) (

85 +

79 )

b) Which of the above fractions is a decimal fraction? 7) If x = - 12, determine the sum of these three numbers:

- 80 – x ; 2x ; and x-0.15

.

8) Find the unknowns angles in the following figure.


20

A

M

D

CB

F

D B

E

C

A

P

Q T

S R

68

112

BF E C

DA

I

H G

9) ABC is right angled at C, M is the midpoint of [AB]. C is joined to M and produced to a point D such that DM = CM. Show that: a) ΔAMC Δ BMD b) ΔDBC is a right angle. c) ΔDBC ΔACB

d) CM = 12

AB

e) (BD)//(AC) & (AD)//(BC).

10) In the figure, (AB) (CD), CST 68 and STE 112. Prove that (AB) (EF). 11) In the figure, (AB) and (DE) are perpendicular to (FC). FB EC and FD AC.

a) Prove that ABC DEF. b) Prove that FBH CEG. c) Prove that HA GD.

d) Prove that HIG is an isosceles triangle. e) Prove that (AD)//(HG).

12) Complete the table then draw a graph for the line y=2x – 1.

x 0 1 2 3

y = 2x – 1 –1 5

13) a) Plot A(5;1) , B(2;1) & C(5;4) in an orthonormal system of axes x'Ox , y'Oy. b) What is the nature of ABC? Calculate its area. c) The // to (BC) draw from A cuts the // to (AB) drawn through C in D. What are the coordinates of D? Calculate ADC . d) Show that BC=AD. e) M & N are respectively the midpoints of [BC] & [AD]. i) What are the coordinates of M & N? ii) Prove that AMB CND.


21

F

E

DA

B

14) Choose, with justification, the correct answer:

1) 2 5 6+3 3 15

in simplest form is ----------

a. 1415

b. 43

c. 3248

2) In the figure, (AB) (CD), (EF) is a transversal. Which of the following must be correct?

I. x p II. x q III. x r a. I only b. I and III only c. I, II and III

3) If 100

97mm

= 8, what is the value of 5

2mm

?

a. 8 b. 3 c. 83

4) If 5 3 y8 4 2 , then y = ?

a. 21 b. 13 c. 40 5) Which symbol makes this statement true? 5(18 – 24) __ 90 - (-120)

a. < b. > c. =

6) In the figure, are BCD and ABD congruent? If yes, state the reason. [Figure may not be drawn to scale.]

a. No b. Yes, A.S.A c. Yes, S.A.S 7) What postulate would you use to prove these two triangles are congruent? Given that DE=BA & (DE)//(BA).

a. SSS b. SAS c. ASA

DC A

B

4546

4645

D

B

E

AC F

qp r

x


22

A

C

B

Translation (Prerequisites)

1) Consider the two segments [AB] & [BC]. a) Draw the line passes through A & // to (BC). Draw the line passes through C & // to (AB). b) The two drawn lines intersect at D. What is the nature of ABCD? ------------------

2) ABC is moved (translated) to another position A'B'C'.

a) Locate C'. b) Compare: AB ----- A'B' AC ----- A'C' BC ----- B'C' c) Complete: (AB)// ----- (AC)// ----- (BC)// ----- d) What is the nature of ABC & A'B'C'? e) Are the triangles ABC & A'B'C' congruent? Why? f) The image of A is -----

The image of B is ----- The image of C is ----- The image of ABC is -----

A B

C

A' B'


23

O H F

C D G

E A B

B

A

Translation 1) Given ABCD is a square of center O, and E, F, G and H are respectively the midpoints of [AB] , [BC] , [CD], and [AD].

a) Given the translate of A is O. Copy & complete: i) The translate of E is ........... ii) The translate of O is ........... iii) The translate of [HE] is ............ iv) The translate of triangle AEO is ............ b) The symmetry of triangle OHD with respect to O is ..............

c) What translation takes O to G? ........... 2) Reproduce the figure then draw its image by the translation from A to B. 3) a) On an orhonormal system of axes x'Ox, y'Oy, plot the points: A(1 ; -1) ; B(2 ; 3) ; C(-2 ; 2) ; D(4 ; 2) & G(0;-4) .

b) Place the point E the image of C by the translation that transforms A to D. c) Place the point F the image of A by the translation that transforms D to B. d) Draw MNP the image of ABD by the translation that transforms O to G. e) How are the segments [AD] & [FB]? What is the nature of CEBF?

f) What are the coordinates of E, F, M, N & P? 4) Draw the image of this figure by the translation of vector

AB :

B

A

B

(d) (C)

()


24

x cm 3 cm

Algebraic Expressions (Prerequisites)

1) Consider the opposite rectangle.

a) Express the area A in terms of x. A = ------------------ b) Express the perimeter P in terms of x. P = ------------------ c) Evaluate A & P for x=2 cm.

2) Calculate:

A= –1 – (– 2) – (– 9) + 15 = ------------------------------------------- B= – 4(– 1 – 8) – 2(– 3 – 6) =------------------------------------------- C= (22 + 33)x(52 –7) = -------------------------------------------

3) Multiply: a) xx =---- b) b7xb4 = ----- c) 3n4x4n2 = ----- d) (-3a2b2)(2ab3) =------

4) Simplify each expression:

a) 2x2 + 5x+ 3x – 5 = ---------------------------------------- b) −2y2 + 3y − y2 − 3y = ---------------------------------------

c) 12 – 4x – 2(5 – 6x) = --------------------------------------- d) 4x + x(2x – 1) = ---------------------------------------

5) Given h(x) = −2x2 − x + 6, find each. Show your work.

h(−3) = --------------------------------------- h(0) = ---------------------------------------


25

R S

T

6) RST is isosceles with base [RS]. RT = 5x – 4, ST = 3x + 4, and RS = 2x + 9,

a) Express the perimeter P in terms of x. P = ----------------------------------------- b) Find P when x=1 cm. P = ----------------------------------------- 7) Complete:

Word Expression Mathematical Expression Sum of x & y Difference between x & y Product of x & y Double x; Twice x Triple x Quotient of x & y Reciprocal of x One fourth of x x subtracted from y x exceeds y by a

8) Write an equation to model the statement.

a) The sum of a number n and twelve is twenty-five. --------------------------- b) Thirty-five is the sum of a number x and eight. --------------------------- c) Three less than a number t is negative sixteen.

---------------------------


26

Algebraic Expressions

1) Complete this table:

Polynomial

Coefficient of 3 2x x x

Constant

term

Degree

a) 3 25x -6x -x +8 b) 3 2-3x + x -18 c) 2x -6x +7 d) x -9

e) 32 x -163

2) a) Fill in the blanks with suitable algebraic expressions. i) 15 10x 5( ) ii) 2pq q q( ) b) Copy & complete:

i) x² – --- + 16 =( --- - ---)2 ii) 1625x²=(4 + ---)(4 ---)

3) Expand & reduce each expression: A= 2(a – b) + 3(a – b) – 4(a – b) ; B= x – [ – 4x – ( – x – 3)] C= (4x –1)(x – 2) ; D= (3x – 2)(2x + 3) E= (2x2 – x – 2)(3x – 1) ; F= (x + 6)2 G= (4x + 3)2 ; H= (2x – 7)2

I= (4x + 3)(4x – 3) –x(4x -1) ; J= (x + 1)² + (x – 3)² 4) Factorize these expressions:

A= 9x2 - 15x ; B=3x3 + 6x2 – 3x ; C= 14x2 + 21x ; D= 21x2 – 7x E=8ab-12bc ; F= 3 26a b +2ab ; G= 7 4 9 2 5 3 9 635a b c -28a b c+21a b c H = 2x(x + 1) + (x – 2)(x + 1) ; I= (7x – 5)(3x + 2) – 6(3x + 2)(x + 3) J= (5x +11)(4y – 1) + (5x +11)(3y + 2) ; K= 2x(x +8) +(x +8)

5) a) Evaluate: 23x -5y -6x +z for x= –1; y= 2 & z= –4. b) Evaluate 3x -10y +135 when x =-5 and y = 2. 6) If (x 6)(x 6) = Ax2 Bx C, find the values of A, B and C. 7) Solve these equations: a) 7x + 19 = –2x + 55 b) 4x=–2(–2x + 3) c) 4(1–x)+3x = –2(x+1)


27

E F

G

10

2x - 3

x + 7

d) 4(3 + 2x) – 6(4x + 3) = 3(x + 3) + 4 e) 4(3y + 1) = 6(2y – 1) + 10

f) - 45

x + 12

= 3 – 710

x g) x2 -

x + 13 +

x - 24 = 0

h) 2x(x – 1) = 2x2 - 4x + 1 i) 3(x+2) - (x-3) =x-5 - 3(x+1) + 4x

8) Given the expression: F = x x +2 - 2x-1 x +2 a) Develop & reduce F. b) Factorize F. c) Calculate F for x 2 . d) Solve the equation F=- x2 .

9) Given: D = - 2x(3x - 5) + (x + 7)(3x - 5). a) Develop & reduce D.

b) Calculate D for x = 53

then for x=0.

c) Factorize D. d) Solve the equation F=-3x2 .

10) Consider the three rectangles: Given the expressions: A = 7(x – 4) ; B = 4(7 – x) ; C = 7(x + 4)

a) Among the expressions A, B & C, which represent the area of the shaded region of figure1, figure2 & figure3? b) Develop A, B & C. c) Calculate the area of the shaded region of each figure for x=5 cm.

d) Calculate x so that the perimeter of figure1 is 71 cm. 11) Given that the perimeter of EFG is 32 cm, a) Write an equation that x satisfies. b) Calculate x. c) Is ΔEFG scalene, isosceles, or equilateral? Justify the answer.

Figure1 x 4

7

Figure2

4

7

x

Figure3

x

7

4


28

12) Given: [AB] and [AC] are the legs of an isosceles ΔABC. m1 = 5x & m3 = 2x + 12 a) Express m2 in terms of x. b) Calculate x. c) Find m2. Deduce mCAB 13) The difference of a number and negative 5 is fifteen. Find the number. 14) Jamil collects cards. Six times the number of cards decreased by 4 is equal to 8 more than three times the number of cards. How many card does Jamil have? 15) Basem bought a book, a magazine and a pen for a total of 14750 LL. The book was 4 times as expensive as the magazine, and the pen was 5500 LL less than the book. Let x be the price of the magazine. a) Label, in terms of x, the price of each article. b) Write an equation. Solve the equation. c) What was the price of each object? 16) Sally is Bob’s younger sister, and she is 7 years younger than Bob. If you multiply Bob’s age by three and subtract Sally’s age, you get their mother’s age, which is 37 years. Let x be Bob’s age. a) Express in terms of x the age of Sally. b) How old is Sally? 17) A cell phone company charges a 29950 LL monthly fee and 120 LL

each minute. If your bill is 51550 LL, how many minutes did you talk this month?

18) The cubes are identical & the balance is at equilibrium. Let x be the mass of one cube in Kg. Calculate the mass of 1000 boxes.

A

B C 3

2 1


29

B

D

C33º

A

M CA

B

6

2

B

D

C(y-10)º

56º

A

Bisectors (Prerequisites)

1) a) What does (d) represent for [IH]? ----------------------------------- b) What does [Ix) represent for GIH? ----------------------------------- 2) a) Draw the bisector of [AB]. b) Draw the bisector of xOy . 3) Given that [BD) bisects ABC .

a) Find ABD . b) Solve for y.

4) Find: BC =-------- AC=---------

60°

H G

I

(d)

60°

x

BA

O

y

x


30

C

A

FB

24

E

H

B

A

C

D

P

Bisectors 1) Consider the triangle ABC where: o o oABK = 70 , BKA = 80 , ACM =25 . (CM) is the bisector of angle ACB . a) Reproduce the figure. b) Calculate the measure of: ACK , BAK , & BAC .

c) What does (AK) represent for BAC ? Justify. d) What is the center of the circle inscribed in ABC? Draw this circle. 2) Given on the opposite figure: (BF) is the bisector of [AC]. (FD) is the bisector of [CE]. a) Reproduce the figure. b) Prove that FA=FE. c) What is the circumcenter of the circumscribed circle about ACE? Draw this circle. 3) ABC is a right triangle at B. [AF) is the bisector of BAC . [AH] is a height. a) Prove that BEF is an isosceles triangle. b) Construct I the circumcenter of the circumscribed circle about ABC. Draw this circle. 4) ΔABC and ΔDBC are two isosceles triangles of same base [BC] and vertices A and D. (AD) is extended to intersect [BC] at P. Show that:

a) (AP) is the perpendicular bisector of [BC]. b) ΔABD ΔACD. c) (AP) bisects BDC .

5) ABC is any triangle. (D) is the parallel to (BC) drawn through A. The bisector of ABC cuts (D) in D. The bisector of ACB cuts (D) in E. a) Draw the figure. b) Prove that each of the triangles BAD & CAE is an isosceles. c) Deduce that: AB+AC =DE. d) Prove that OBAC ABC ACB 180 .

B C

A

M

K

F

ECD

B

A


31

(d)

A C

4cm

B

Locus (Prerequisites)

1) Draw these figures:

a) ABC isosceles of vertex A. b) MNP right at N. c) Circle C(O; 2 cm).

2) a) Draw the point A at distance 2 cm from (d). b) Draw the line (d')//(d) through A.

3) a) Draw (d) the bisector of [AB]. b) Locate the point C on (d) where AC= 3 cm. c) Calculate BC. BC=------------ 4) a) What is the center M of the circumscribed circle (C) about ABC? ------------------------------------------- b) Draw (C). c) Calculate the radius of (C). R=-----------

5) What is the path of these objects? a) A dark spot on the arm of a running fan? ---------- b) A car moves on a main street? ------------ c) The end point of the minute's arm of a clock? -----------------

BA


32

Locus 1) Triangle ABC is an isosceles triangle of variable vertex A. [BM] is the height relative to [AC]. a) Determine the locus of A. Draw this locus. b) Find the locus of M. Draw this locus. 2) Given a circle (C) of center O. Let A be a fixed point on (C) and E be a variable point on (C). M is the midpoint of [AE]. a) Prove that (OM) is perpendicular to (AE). b) Find the locus of M as E describes (C). Draw this locus. 3) Consider the circle (C) of variable center O. [AB] is a fixed chord. Determine the locus of O. Draw this locus. 4) Given the circle C(O;3cm). M is a variable point on (C). I is the midpoint of [OM]. a) What is the locus of I? Draw this locus. b) E is a point outside (C) at distance 1 cm from (C). What is the locus of E? Draw this locus. 5) Two dogs are tied to posts A and B in a rectangular field as shown. Each dog can reach 6 meters from its post. a) Draw the diagram of scale: 1 m 1 cm. b) What is the locus of each dog?

Draw the loci to show the area which both dogs (a) & (b) can reach.

c) Can the dogs catch each other?

10 m

10 m

8 m

4 m

4 m

A

B

(b)

(a)


33

Family members1 2 3 4 5

1

2

5

Num

ber

of s

tude

nts

Statistics (Prerequisites)

1) The line graph shows the average temperature during a week.

a) What was the temperature on Tuesday? ---------------- b) What was the highest temperature of the week? ------------ c) When was the temperature 26C? ---------------- 2) The given below are the numbers of family members of 30 students in a class. 2 3 4 1 5 4 3 4 5 2 1 2 3 5 3 2 5 2 3 5 2 5 5 5 4 5 5 4 3 4

a) Complete the table: b) Complete this bar graph.

Family members 1 2 3 4 5 Total Number of students 2 6

18

20

22

24

26

28

Mon Tue Wed Thur Fri Sat Sun

Tem

pera

ture

(C)


34

Number of goals

Num

ber

of m

atch

es

0 1 2 3 4 5

5

8

12 11

6

Statistics

1) The list below indicates the grades of 25 students of the 7th grade in the mathematics exam.

a) Copy & complete the table:

b) Represent the freq. by a bar graph & the % rel. freq. by a polygon graph. c) How many students took a grade: i) less than 12? ii) at least 12? iii) at most10? iv) more than 10? 2) Some codes of telephones in Lebanon are: 01, 02, 03, 04 and 05. In an enterprise which have done 1500 calls, we have registered the table aside: a) Find the number of calls done in Beirut whose code is 01. b) What is the percentage of calls done in the 02 region? 3) During a football league, the number of goals scored per match is given by the following graph: a) What is the number of matches played during this season? b) Give a frequency & % rel. freq. distribution table for this diagram. c) Find the number of matches such that at least two goals were scored in each. Find their percentage. 4) The opposite chart shows the methods used by 60 different students to get to school.[AB] is a diameter & O is the center. a) What is the % of students that come by each method? b) Calculate the number of students that use each method.

7 10 12 15 15 10 7 7 7 15 10 7 12 7 7 10 10 7 10 10 12 12 7 15 12

Grades Total Frequency Relative frequencies Relative frequencies in %

Code Number of calls

01 02 03 04 05

x 330 144 261 171

walk

car

bus

A

O

B

60°


35

Proportions (Prerequisites)

1) Which pair of ratios is a proportion?

a) 1 3 = 2 6

b) 15 72 = 24 45

c) 14 42 = 18 54

2) A giant tortoise takes about 20 minutes to travel a distance of 100 meters.

a) Complete this table to show how long it takes to travel other distances (assuming it always moves at the same constant speed):

Distance traveled (in meter) 100 200 300 400 500

Time taken (in minutes) 20

b) How long would it take the tortoise to travel 250 meters? ------------------------------------- c) How far does the tortoise travel in 50 minutes?

------------------------------------- 3) Find the value of x in each of the following proportions.

a) 1 x=4 28

b) 8 18=x 9


36

Proportions

1) Which is a proportionality table?

a) b)

2) Complete this proportionality table:

5 25 b c 7 a 63 2.1

3) A car consumes 6.8 liters of fuel in 100 km. a) How many liters this car consumes in 275 km? b) How many km this car travels with 51 liters of fuel? 4) On a map of Lebanon, 1½ cm represents 50 km. If it is approximately 2½ cm from Tyr to Beirut on the map, what is the actual distance in km? 5) A pastry shop sells cakes where the price of one cake is 500 LL (All cakes have the same price). a) Copy & complete this table: b) Are the quantities x & y proportional? Justify the answer. c) On an orthonormal system of axes x'Ox & y'Oy, draw points (x;y) Use the scale: On x-axis: 1 cm 100 cake On y-axis: 1 cm 100,000 LL. d) Answer these questions graphically by making the necessary drawings: i) Calculate the price of 250 cakes. ii) Calculate the number of cakes whose price is 850,000 LL.

8 16 800 2.4 4 8 400 1.2

9 2.4 900 36 3 0.8 301 12

Number of cakes (x) 100 800 Total price in LL (y) 600,000


37

2 cm

2 cm

4 cm

2 cm

8 cm

6 cm

10 c m

Space Figures (Prerequisites)

1) Consider these figures:

Complete this table: Type Number of vertices Number of edges Number of faces Fig1 Fig2 Fig3 2) Find the area & perimeter of each of the following figures.

a) b) A=---------------- A=---------------- P=----------------- P=----------------- c) d) A=---------------- A=---------------- P=----------------- P=-----------------

C B

G

H

F

D

E

A

Fig1

D

E F

G H

A

B C

Fig2

A

B

C

D E

F

Fig3

18cm


38

2 c m 2 c m

2 cm

3 cm 4 cm

3 cm 4 c m 6 cm

1 cm

5 cm

O

15 cm

Space Figures 1) Find the volume and total surface area of each of the following prisms.

a) b) c)

2) Find the volume of each of the following solids. a) b)

3) The figure shows a cylindrical glass with half glass of water. Find the volume of water. 4) The length, width and height of a room are 8 m, 6 m and 2.5 m respectively.

The total area of a door and a window occupies 15

of the area of the walls in the

room. If the ceiling and the walls are to be painted, a) find the total painted area. b) Given that the capacity of a tin of paint is 2 L and each L of paint can paint an area of 6 m2, how many tins of paint are required?

20 cm

Area 12 cm 2

12 cm

Area 8 cm 2


39

C

A B

x - 3 2x + 4

F

D E 2x - 17

26

2x - 3

x +

1

x - 1 2(x – 1)

x – 1

Final Review 1) Expand & reduce:

A=(-6x – 15)(3x – 5) ; B=(2x − 5)(x − 2) − (6 − 9x) + x2 ; C=(3 – x)² + (x + 5)² 2) Factorize: A=30x – 42 ; B= 3 7 8 2 5 4 2 6 7-36x y z -12x y z -48x y z ; C=4x(x-7)-3y(x-7) 3) Given that T (x2y 60)(1 2xyn), x -3, y 4 and n 2, find the value of T. 4) Solve these equations:

a) 14

(12x + 16) = 10 – 3(x – 2) b) (2x2 – x + 6) – 2(x2 – 3x + 5) = 11 + 5x

c) 3x – 2(x – 4) = 5 + 4(1 – 2 x)

y 1 2y 3 1d) 4 3 2

e) 5(-2x -1) =-8x -5

5) Given the polynomial: C = (x – 1)(2x + 3) + (x – 1)².

a) Show that C = 3x² – x – 2 . b) Evaluate C for x = -2. c) Factorize C.

d) Solve the equation C= 3x². 6) Given: A = (x – 3) (x + 3) – 2(x –3) & B = x(x+1) + 3(x+1)

a) Factorize A & B. b) Develop & reduce A & B. c) Evaluate A for x = -1 & B for x = 0.

d) Solve the equations: i) A=x² ii) A=B 7) Given ABC DEF. a) Find the value of x.

b) Deduce AB & DF. 8) The opposite figure consists of a rectangle, square & a right triangle. a) Express, in terms of x, the area of the: rectangle, square, triangle, shaded surface. (write the answers in developed form) b) Calculate x so that colored area is 22 cm². c) Calculate x so that perimeter of the rectangle is 20 cm.


40

9) The price of a pair of shoes is double the price of a pair of trousers. The price of a hat is 20000 LL less than that of a pair of trousers. The total cost of all above objects is 176000 LL.

The above given is translated by the equation:

x + 2x + x – 20000 = 176000 a) What price does x represent?

b) Solve the given equation. c) Calculate the price of each article. 10) Showing all steps, calculate & write the result in simplest form.

3 16 64A4 5 15

;

5 2( 7) 11B

-2 - 6 3 + 5 ; C =

4 (10-2)3 102

12 10-3

11) The following shows the sale of juice in a store.

A O W A O O M O A W M O O A M W M A M A O A A W A O A M W O W A O O A O M O A O M O W O A O A A O A

A: Apple juice W: Watermelon juice O: Orange juice M: Mango juice

a) Organize the data by using a frequency distribution table. b) Which one was the most popular juice? c) What is the percentage of selling Mango juice?

d) Make a frequency bar graph & represent the % rel. freq. by a polygon graph. 12) ABC is a right isosceles at A. The to (AB) at B cuts the to (AC) at C in I. a) Draw the figure. What is the nature of ABIC? b) Prove that (AI) is the perpendicular bisector of [BC]. c) M is on [AB] & N on (AC) where A, C & N are in this order & CN = BM. i) Prove that the triangles IBM & ICN are congruent. ii) Deduce that the perpendicular bisector of [MN] passes through I. 13) a) On an orhonormal system of axes x'Ox, y'Oy, plot the points: A(-1 ; 1) ; B(-4 ; 1) ; & C(-1 ; 3) . b) Calculate the area of ABC. c) What are the coordinates of the center I of the circumscribed circle (C) about ABC? Draw (C). d) Draw the images A'B'C' & (C') of ABC & (C) by the translation that transforms B to A. What are the coordinates of A', B', & C'?


41

1 2

F60

C

A

E

BD

Q P 4x 2x

B

M N

CA1

23

14) The height (in m) of a tree relative to its age (in years) is tabulated as follows: a) Are the age of the tree & its height proportional? Justify the answer. b) Represent graphically the points (x;y). c) Use the graph to calculate the height of the tree after 7 years. d) After how many years the height of the tree will be 50m? 15) ABC is a right isosceles triangle at A. I is the midpoint of [BC]. L & K are two points where AL=KC. a) What is the nature of AIC? Justify the answer. b) i) Show that LAI IKC. ii) Deduce that 1 & 2 are complementary. iii) What is the nature of LIK? Justify the answer. c) Draw the image L'B'I' of LBI by the translation that transforms A to I. 16) ABC is a triangle where AB=6 cm, A =110° & B =15° . The bisector of A cuts [BC] in D.

a) Draw the figure. b) Prove that ADC is an isosceles triangle. c) The bisectors of [AB] & [AC] intersect at E. Prove that IBC is an isosceles triangle.

d) Construct the circumscribed circle about triangle ABC. 17) Prove that (AB)(CD).

18) In the figure, (AM)//(BN), (MB)//(NC), AM BN and MB NC.

a) Prove that AMB BNC. b) i) Prove that 1+2+3 =180° . ii) What do you can say about the points A, B, & C?

c) Show that B is the midpoint of [AC]. d) Show that (MN)//(AC).

Age (x) 1 2 3 4 5 Height (y) 0.5 1 1.5 2 2.5


42

A

J K

D C

I B 8 cm

x 10 cm

H

R Q P O

J K L M N

I G F

A B C D E

19) Choose, with justification, the correct answer:

1) Which expression cannot be factorized?

a. ab + bc b. 2x + 6y c. x + 2y

2) Six times the sum of a number and 12 gives a result of 22. The number can be found by writing and solving the equation ----------

a. x + 12 = 226 b. 6(x + 12) = 22 c. 6x + 12 = 22

3) If we subtract 4x2 + x - 2 from 3x2 – 2x + 9, the result is --------

a. -x2 – 3x + 11 b. x2 + 3x – 11 c. 7x2 – x + 7

4) ABCD is a square, AIJK is a rectangle. Calculate x so that the area of AIJK is equal to half the area of ABCD.

a. 6.25 cm b. 50 cm c. 5 cm

5) [AB] is a segment. The locus of Q that moves in such a way that it is always equidistant from A and B is the ------

a. bisector of [QA] b. bisector of [AB] c. C(A;AQ)

6) If k 110 =2

, find the value of k+310 is:

a. 72

b. 500

c. 1003

7) By the translation that transforms A to B, the image of JOPK is ------

a. KPRM b. KLQO c. KPQL

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