congruency algebra expression sec 2

7/29/2019 Congruency Algebra Expression Sec 2

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1. In the diagram below, MNLABC . Find the values of the unknown marked.

Answer: x = .[1]

y = .[3]

z= .[1]

2.The curved surface area of a cone of radius r and height h is given by 22 rhrV . Make h thesubject of the formula.

Answer: [3]


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3. Find x and y if the triangles are congruent.

Answer: x = .

[2]

y = cm [1]

4.Given that

2

)3(

gmA ,

(a) expressgin terms ofA and m,

(b) find the value ofgwhenA = 60 and m = 12.

Answer: (a) [2]

(b) [1]


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5.(a) Simplify the following expressions ,

22

yx

yx

(b) Factorise 32 805 yyx

(c) Solve the equation 812)13( 2 xx

Answer: (a) [2]

(b) [2]

(c) [3]


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6. Make y the subject of the formula .53

322

y

yxx

Ans: ___________________ [2]

7. In the diagram below, ABCis similar to ADE.AE= 2 cm,BE= 7 cm,DE= 4 cm,

BC= 12 cm,DC= 3 cm, EAD = 104 and ABC= 29.

(a) Find AED,

(b) Calculate the length of AD.

Ans: (a) AED = ____________ [1]

(b) AD = ____________ cm [2]

B

A

E

D

C

104

2 cm

7 cm

3 cm

4 cm

12 cm29

?


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8. Solve the following equations:

(a) 34

2

x

x.

(b) .25

8

x

x

Ans: (a) __________________ [2]

(b) __________________ [2]


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9. Simplify the following:

(a)32

4

3

2

49

21

8

3

cab

ac

ab

ba

(b)4

12

2

32

xx

Ans: (a) __________________ [1]

(b) __________________ [3]

10. In the diagram,PQR and QSTare straight lines. It is given that TQ is perpendicular toPR,PT=RS,

QR = 2QS, where Sis the midpoint ofQTand .280QRS

(a) Name a pair of congruent triangles .

(b) If the length ofQSis 20 units, find the ratioPQ : QR.

Ans: (a) __________________ [3]

__________________

(b) __________________ [2]

P Q

S

R

T


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11. Factorise )()( 3223 bbaaba .

Ans: ____________________ [3]

12. Given that2

2

3

b

ba, express b in terms ofa.

Ans: [3]


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13. Simplify1

3322

x

x

xx

x.

Ans: [3]

14. Triangles PQRand STRare similar. Find the length ofPS.

Ans: cm [2]

2 cm

5 cmPR

4.5 cm

Q

S

T


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15. The diagram below shows a quadrilateralABCD. The diagonalsACandBD

intersect at T.AT=BTand CT=DT.

(i) Name a triangle which is congruent to triangleATD. Give your reasons

and state the case of congruence.

[3]

(ii) Name a triangle which is similar to triangle CDT. [1]

(iii) Given that DAC=x and CAB =y, express ACB in terms ofx and

y.

[2]

(iv) Given also thatAB = 3 cm,AT= 2 cm andDB = 7 cm, calculate the

length ofDC.

[2]

A

C

B

D

T

xy


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16.a) Simplify

82

2322

2

R

RR b) MakeA the subject of the formula

DA

RAD

2

[3]

[3]


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17.(a) Expand the expression

2

52

y

x

(b) Make a the subject of the given formulae:

a

bacx )(

(c) Express the following as a single fraction in the simplest form:

mn

mm

nm

mm

232 623

[2]

[2]

[2]


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18. Express, as a single fraction, in its simplest form:

xx 615

3

52

5

Ans: [2]

19. (a) Solve the following equation.

2

1

3

1)52(

4

1 xx

(b) Given that1

232

x

xy , make xthe subject.

Ans: (a) [2]


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(b) [2]

20.

In the diagram above, triangleABCis enlarged to triangleAPQ.AC=x cm,AB = 4 cm,BP=y cm, CQ = 8 cm,BC= 6 cm andPQ = 15 cm. LineBCis parallel to linePQ.(a) Find the value ofx.(b) Find the value ofy.(c) Find the scale of enlargement, k, from triangleABCto triangleAPQ.

Ans: (a) cm [2]

(b) cm [2]

A

B C

P Q

x cm

8cmy cm

4cm

6cm

15cm


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(c) [1]

21. Factorise completely a2b29a2.

Ans: _____________________[2]

22. (i) Given that xy

y

1, expressy in terms ofx.

(ii) Hence or otherwise, find the value ofy whenx =2

1 .

Ans: (i) _______________________[2]

(ii) _______________________[1]


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23. The product of two consecutive even numbers is 48. If the smaller number isx,

write down a quadratic equation inx. Hence solve it to find the smaller number.

Ans: ______________________[1]

______________________[2]

24. (i) Factorise completely 2x2 + 5x3.

(ii) Solve 2x (2x + 5) = 6.

Ans: (i) _______________________[1]


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(ii) _______________________[2]

25. In the diagram, the right-angled triangle ABC is similar to triangle ADE, whereAD = 9 cm, BD = 6 cm and AC = 5 cm.Find(i) BC,(ii) area of triangle ADE.

A

5 cm

B C

6 cm

D E

Ans: (i) ____________________[1]

(ii) ____________________[2]

26. Expand (xy)(x +y). Hence evaluate )2

12)(

2

12( .


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Ans: _______________________[2]

27. (a) Expressxx 2

1

)1(3

2

as a single fraction in its simplest form.

(b) Simplify2

2

)2(4 x

xy

x

xyx

.

Ans: (a) ______________________[2]


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(b) ______________________[3]

28. Solve the following equations:

(a) xx 74 2 (b) 2053 xx

(a). ________________ [2]

(b). ________________ [3]

29. In the diagrams below, triangle ABC is similar to triangle LMN.

Find the unknown anglex and sidey marked in the diagrams.

x =______________ [1]

L

M

N

4 cm

3 cm

x

y cm

7 cm

C

B

A

40

55


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y =____________cm [2]

30. Steven boughtx T-shirts costing a total of $180 from a wholesaler.

(a)Write an expression, in terms ofx, the cost price of one T-shirt.

(b)If he wants to sell the T-shirts at a profit of $3 for each T-shirt, write an

expression, in terms ofx, for the selling price of each T-shirt.

(c)He sold 28 T-shirts at a profit of $3 each while the rest were sold at $7 each.Write an expression, in terms ofx, for the total amount of money he wouldreceive for selling all the T-shirts.

(d)If he made a profit of $100 for selling all the T-shirts, form an equation in

terms ofx and show that it reduces to .0720562 xx [2]

(e)Hence find the number of T-shirts Steven bought.

(a). $_________________ [1]

(b). $_________________ [1]


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(c). $_________________ [2]

(e). ___________ T-shirts [2]

31. In the diagram below, PQ is parallel to BC.

Given that AQ = 6 cm, PQ = 12 cm, AB = 20 cm, BC = 16 cm, AP = x cm and QC =

y cm.

(a) Name the pair of similar triangles in the figure(b) Findx andy

(a). _____similar to _____ [1]

16

Q

P

A

B

C

12y

20x

6


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(b).x = _____________cm [2]

y = ______________cm [2]

32. Simplify each of the following:

(a)bybxayax

bxax

1215810

302022

Answer [2]

(b)2

11

149

132

xxx


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Answer [2]33. In the figure below, PMNis similar to PQR. Find the value ofPNandNQ.

Answer PN= . cm [1]

NQ= cm [2]

34. In the figure below, AED is enlarged to ABC. Given thatAED =ABC,AD = 3 cm,AE= 5 cmandEC= 2 cm, calculate

(a) the scale factor of the enlargement.(b) the length ofAB.

MP

N

Q

R

5 cm

15 cm

9 cm 4 cm

B

D

3 cm

A E C5 cm 2 cm


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Answer (a) [1]

(b)AB= .. cm [1]

35. Evaluate2

4

11

222

xx

x

x

x

x.

Answer [4]

36. If ttrs 24 , express tin terms ofrands in its simplest form.


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Answer [3]

7. Factorize the following completely

(a) 21112 2 xx ,

(b) 72y2 50x2,

(c) 4p(rs) + (p + q) (2s2r).

Answer (a) [1]

(b) [2]

(c) [3]

38. Simplify the following algebraic fractions. Leave your answer as a fraction wherevernecessary.

(a)22

dc

bcadbdac

[2]

(b)

a

b

b

a

b

a

a

b

[2]


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39. Makex the subject of the formula2

2

5

32

x

xy

[3]

40. Find the two consecutive negative odd numbers such that the square of their sum exceeds thesum of their squares by 70. Use ONLY algebraic method to solve. [5]


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In the figure below, BACis similar to CED. Given that AC= 10 cm, DE= 6 cm,

AB= 4 cm and DCE = 90o. Find(a) ABC(b) the length ofEC.

(c) DCBC:

Answer: (a) [1]

(b) cm [2]

(c) [1]

2. In the diagram below, triangleABCis congruent to triangleDBE,AC= 13 cm,BC= 5 cm andDB = 12 cm. Findthe length of

(a) DE,(b) AE.

A

B C

D

E

4 cm

6 cm10 cm

A

DB

C

E

5 cm12 cm

13 cm


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43. In the diagram below (not drawn to scale), BCis parallel to DEF, CGEis parallel to ABD, BD= 16

cm, DF= 28 cm and BC= 21 cm.

(a) Name two triangles which are similar to BDF. [2]

(b) Calculate the lengths of

(i) EG, [2] (ii)

AB. [3]

Ans; (a) ___________ cm [1]

(b) ___________ cm [2]

A

B

C

D

E

F

G

16

21

28


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44. (a) In the diagram, WXYand WQPare similar,PX 4 cm, WX 12 cm,PQ 21 cm,

WY 6 cm and 119WPQ . Find

(i) the length of XY, [2]

(ii) the length of QY, [2]

(iii) XYQ . [1]


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45. In the diagram, ABCis congruent to XYZ. AB = 8.7 cm,BC= 6.5 cm, CZ= 8.2 cm and AY= 2 cm.Find the length of

(a) AZ,(b) XZ.

Ans: (a) cm [1](b) cm [1]

46.Given that

2

13

p

pq ,

(i) express p in terms of q , giving your answer in its simplest form. [2]

(ii) calculate the value of p when q 6. [1]

Y2

8.7

8.26.5

A

B

C

Z

X


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47. The diagram shows triangle ABC. PQ is parallel to BC. AQ = 12 cm, PB = 6 cm,QC = 4cm and BC = 12 cm.

(i) Prove that ABC is similar to APQ. [2]

(ii) Calculate the length of PQ. [1]

(iii) Calculate the length of AB. [1]

C

A

B

P

Q

12

6

4

12


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48. In the diagram, ABC is congruent to XYZ. The dimensions given are in cm.Find

(a) the length ofAC,(b) the length ofYZ,(c) YXZ,(d) ACB.

(a) [1](b) [1](c) [1](d) [2]

49. In the diagram, DEFis similar to XYZ.

Find(i) XYZ ,

(ii) the length ofDF.

Answer:

D

E

F

X

Y

Z85

16 cm 12 cm

18 cm


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(b) (i) XYZ = [1]

(ii) DF= cm [2]50. Given that ABC is similar to APQ , AB = 8 cm, BC = 6 cm, AC = 7 cm,

CQ = 21 cm, BP = x cm and PQ = y cm, calculate the value of x and y .

Ans: ___________cm

[3]y __________ cm

x

P Q

B C

A

8cm 7cm

6cm

y

21cm


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51. In the diagram, ABC and DEC are similar. Given that CE = 3 cm, CD = 5 cmand DB = 2 cm, find the length AE.

Answer: AE =___________cm [3]

congruency algebra expression sec 2

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