Download - 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]