M_BANK\YR12-2U\APPLICATIONSOFGEOMETRICALPROPERTIES.CAT
Applications of Geometrical Properties
1)!
2U84-2i
Find the area of the rhombus ABCD given AB = 10 cm and EB = 8 cm.
10 cm
A
B
8 cm
E
D
C
†
« 96 cm2 »
2)!
2U84-8i
W
Y
Z
V
3)!
NOT TO
SCALE
X
In the triangle WXV, YZ = 9 cm, VX = 12 cm, WX = 8 cm and YZ || VX. Prove that  WZY is
similar to  WXV and find the length of WZ.†
« 6 cm »
2U84-9iv
A
NOT TO
SCALE
C
B
D
F
4)!
E
In the figure, AB = AC, BD || FE, BF || DE and  CAB = 54. Find the size of  FED
giving reasons.†
« 63 »
2U85-7i
P
Q
T
NOT TO
SCALE
S
R
PQRS is a square with PQ = 1 unit. Find the perimeter of PTRS.†
« 4  2 units »
5)!
2U85-7ii
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A
Z
D
X
Y
NOT TO
SCALE
C
B
ABC and ABD are two triangles, X, Y and Z are points such that XY CB and YZ BD. Prove
that XY : YZ = CB : BD.†
« Proof »
6)!
2U86-5i
List three properties of a rhombus.†
« All sides are equal. Opposite sides are parallel. Diagonals bisect each other at right angles. The
diagonals bisect the angles through which they pass. »
7)!
2U86-5ii
A, B, C are collinear points. BD || AE, BA || DE, BC = BD and  BCD = 58. Reproduce
this diagram on your answer sheet and find the size of  DEA.
C
58
B
A
8)!
NOT TO
SCALE
D
E
†
« 116 »
2U86-5iii
In the triangle PSU, QR || SU, SP || TR, ST = 7·5 cm, PQ = 10 cm, PR = 12 cm and
UT = 15 cm. Find the length of SQ giving reasons.
P
Q
NOT TO
SCALE
R
S
T
U
« 20 cm »
9)!
2U86-5iv
GL is a median in  HFG and HJ || FK.
a.
Draw a neat sketch of this diagram on your answer sheet.
b.
Prove, giving reasons, that KL = LJ.
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F
NOT TO
SCALE
K
L
J
H
10)!
G
« Proof »
2U87-5i
In the diagram AB || CD and GH  AB. If y = 25 find the size of  GMH. Hence or
otherwise find the size of  MFD.†
y
G
A
H
M
C
11)!
B
F
«  GMH = 65,  MFD = 115 »
2U87-5ii
PQRS is a trapezium with PQ || SR. Diagonals PR and SQ intersect at T.
P
Q
NOT TO
SCALE
T
12)!
D
S
R
a.
Reproduce this diagram on your answer sheet.
b.
Prove, giving reasons, that  PQT |||  RST.
c.
Hence, find PQ, given that SR = 36 cm, PT = 5 cm and RT = 15 cm.†
« a) Proof b) 12 cm »
2U87-5iii
In the diagram below,  UXY =  UYX and XZ = YZ.
U
X
Y
Z
a.
b.
c.
V
W
Copy this diagram on your answer sheet.
Prove that  UVY   UWX, giving reasons.
Hence prove that  VZW is isosceles.†
« Proof »
13)!
2U88-6i
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F
14)!
I
E
G
J
H
In the figures FG = 10 cm, EG = 15 cm, EF = 12·5 cm, IJ = 4 cm, HJ = 6 cm and
HI = 5 cm.
a.
Draw a neat sketch and mark on it all the given information.
b.
Show that  EFG |||  HIJ giving reasons.†
« Proof »
2U88-6ii
L
M
P
15)!
NOT TO
SCALE
NOT TO
SCALE
K
N
The figure above shows quadrilateral KLMN with diagonals KM and LN intersecting at P.
a.
Reproduce this diagram on your answer sheet.
b.
If the diagonals KM an LN bisect each other at right angles, prove that KLMN is a
rhombus.†
« Proof »
2U89-3b
L
NOT TO
SCALE
N
K
M
In the diagram above, KN = NM, KL = LM,  KNM = 110 and
 NKL = 45.
i.
Reproduce a neat sketch and mark on it all the given information.
ii.
Find the size of  MKN and  KLM, giving reasons.†
L
N
45 110
M ii)  MKN = 35,  KLM = 20 »
« i) K
16)!
2U89-6c
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P
Q
R
U
T
17)!
NOT TO
SCALE
S
In the diagram above, SQ  PQ, RU  SQ and PS || QR.
i.
Prove that  RQU |||  PSQ.
ii.
If RU = x units, QR = y units and PS is four times the length of RU, find the length
of PQ in terms of x and y.†
4x2
« i) Proof ii)
»
y
2U90-2c
L
NOT TO
SCALE
N
M
K
 KLM is an isosceles triangle with KL = LM,  LKM = 80, LN bisects  KLM and
 KMN = 20.
i.
On your answer sheet, draw a neat sketch of the diagram above, showing all the
given information.
ii.
Find the size of  LMN, giving reasons for your answer.
iii.
Find the size of  LNM, giving reasons.†
L
N
80
20
ii)  LMN = 60 iii)  LNM = 110 »
« i) K
18)!
M
2U90-5d
PQRS is a quadrilateral with PR = QS, PQ  PS and SR  PS.
i.
On your answer sheet, draw a neat sketch and mark on it all the given information.
ii.
Prove that  QPS and  RSP are congruent.
iii.
Hence prove that PQRS is a parallelogram.†
P
Q
« i) S
19)!
2U91-4c
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R ii) iii) Proof »
M_BANK\YR12-2U\APPLICATIONSOFGEOMETRICALPROPERTIES.CAT
In the diagram given below,  ABC is a right angle triangle with  BAC = 90, CQ = CR,
PB = RB and  ACB = 40.
A
P
i.
ii.
20)!
Q
40°
B
R
C
Copy this diagram onto your answer booklet.
Write down the size of  PRQ. (No reasons are required in your solution).†
« 45 »
2U91-7a
P
Q
V
21)!
T
S
R
PQRS is a parallelogram. TQ bisects  PQR and VS bisects  PSR.
i.
Copy this diagram onto your answer booklet.
ii.
State why  PQR =  PSR.
iii.
Prove that  PVS and  RTQ are congruent.
iv.
Hence find the length of TV if PR = 20 cm and TR = 8 cm.†
« ii) Opposite 's in a parallelogram iii) Proof iv) 4 cm »
2U92-5a
N
L
123
K 2 
22)!
NOT TO
SCALE

M
NOT TO
SCALE
J
In the diagram above JKLM is a quadrilateral and LMN is a triangle. JM || LN, JK = KL,
JM = ML = MN,  KLM = 123,  JKL = 2 and  JML = .
i.
Copy this diagram onto your answer sheet.
ii.
Show that  JML = 38 giving reasons.
iii.
Determine the size of  LNM giving reasons.†
« ii) Proof iii) 38 »
2U92-7a
P
NOT TO
Q
SCALE
N
M
Z
S
R
In the given diagram PQ || RS. MQ bisects  PQR, NR bisects  QRS and MQ = NR.
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i.
ii.
iii.
iv.
Copy this diagram onto your answer sheet and mark on it all the given information.
Explain how you know that  MQZ =  NRZ.
Prove that  QMZ   RNZ.
Hence prove that the intervals QR and MN bisect each other.†
P
Q
N
M
Z
R
« i)
23)!
S ii) iii) iv) Proof »
2U93-4a
In the diagram CA = AD = DB and  EBD = 20. Copy this diagram onto your answer
sheet.
C
NOT TO
SCALE
D
i.
ii.
20
E
A
Show  ADC = 40, giving reasons.
Hence find the size of  CAE, giving reasons.†
B
« i) Proof ii) 60 »
24)!
2U93-5c
In the diagram CT bisects  ACB, AE is perpendicular to CT and M is the midpoint of AB.
AE produced meets BC at the point P.
C
P
E
i.
ii.
iii.
iv.
A
T M
B
Copy this diagram onto your answer sheet and mark in all the given information.
Prove that  ACE is congruent to  PCE.
Explain why AE = EP.
Hence prove that EM is parallel to PB.†
C
P
E
A
T M
B
« i)
ii) Proof iii) Corresponding sides in congruent 's iv) Proof »
25)! 2U94-3b
In the diagram AE || BD, AC || ED,  AED = 130 and  ABC = 90.
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A
B
C NOT TO
SCALE
130
E
i.
ii.
26)!
D
Copy this diagram onto your answer sheet.
Find the size of  BAC giving reasons.†
« 40 »
2U94-7c
In the figure triangles ACB and APO are equilateral.
A
P
NOT TO
SCALE
O
i.
ii.
iii.
iv.
B
C
Copy this diagram onto your answer sheet and include all the given information.
Explain why  BAO =  PAC.
Prove  AOB   APC.
Hence prove OB = CP.†
A
P
O
27)!
C ii) Each angle is equal to 60   OAC iii) iv) Proof »
« i) B
2U95-1d
In the diagram AB || CE,  ABF = 75 and  BFE = 35.
A
C
NOT TO
SCALE
B
75
F
35
D

E
Find the size of  giving reasons.†
« 40 »
28)!
2U95-5d
E
2y
F
y
L
96
G
H
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The diagram shows a rhombus EFGH. A line EL is drawn through E so that
 HEL = 2   FEL.
i.
Copy the diagram onto your answer page.
ii.
 FGH = 96, find the size of  ELF giving reasons.†
« 106 »
29)!
2U95-9b
In the diagram ABCD is a square. AB is produced to E so that AB = BE and BC is produced
to F so that BC = CF.
A
B
E
D
i.
ii.
iii.
NOT TO
SCALE
C
F
Copy the diagram onto your answer page.
Prove  AED   BFA.
Hence prove  AED =  BFA.†
« Proof »
30)!
2U96-1f
D
A x
NOT TO
SCALE
30
35
B
50
C
E
Find the value of x.†
« 115 »
31)!
2U96-2b
In the diagram, AB || CD, AD = CD and  BAC = 120. Copy the diagram onto your answer
sheet.
A
B
120
C
i.
ii.
32)!
D
NOT TO
SCALE
Explain why  ACD = 60.
Show that  ADC is equilateral, giving reasons.†
« Proof »
2U96-9a
 ABC is right-angled at A and AD is drawn perpendicular to BC. AB = 15 cm and
AD = 12 cm. Copy the given diagram onto your answer sheet.
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B
NOT TO
SCALE
15 cm
12 cm D
A
i.
ii.
iii.
C
Show that BD = 9 cm.
Prove that  ABC is similar to  DBA.
Hence find the length of AC.†
« i) ii) Proof iii) 20 cm »
33)!
2U97-3b
A
NOT TO
SCALE
B
C
D
ABC is an equilateral triangle. BC is produced to D so that BC = CD.
i.
Copy the diagram onto your answer sheet and mark on it all given information.
ii.
Prove that  BAD = 90.†
A
60
60
« i) B
34)!
60
C
D ii) Proof »
2U97-10a
A
12 cm
B
NOT TO
SCALE
9 cm
M
35)!
D
C
ABCD is a rectangle with AB = 12 cm, AD = 9 cm and AM is perpendicular to BD.
i.
Copy the diagram onto your answer sheet.
ii.
Find the length of BD.
iii.
Prove that  ABM is similar to  DBA.
iv.
Hence find the length of BM.†
« ii) 15 cm iii) Proof iv) 9·6 cm »
2U98-1e
T
y
126
P Q
R S
In the diagram PQT = 126 and  QTR = 90. Find the value of y.†
« 144 »
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36)!
2U98-5c
M
37)!
R
P
O
PN is a diagonal of the rectangle MNOP. R is the point on PO and  PQR = 90.
i.
Prove that  PQR is similar to  NMP.
ii.
Given MP = 5 cm, MN = 10 cm and QR = 2 cm, find the length of PQ.†
« i) Proof ii) 4 cm »
2U99-1d
Q
85
NOT TO
P 120
SCALE
S
38)!
N
Q
75

R
T
In the diagram, PQ || TR,  PQR = 85,  QPS = 120,  PSR = 75 and  SRT = .
Copy the diagram onto your answer sheet. Find the value of .†
« 15 »
2U99-4d
W
X
NOT TO
SCALE
P Q
Z
39)!
Y
WXYZ is a parallelogram. XP bisects  WXY and ZQ bisects  WZY. Copy the diagram onto
your answer sheet.
i.
Explain why  WXY =  WZY.
ii.
Prove  WXP is congruent to  YZQ.
iii.
Hence find the length of PQ given WY = 20 cm and QY = 8 cm.†
« i) ii) Proof iii) 4 cm »
2U99-7a
NOT TO
Y
SCALE
8 cm
P
X
6 cm
Q
R
S
In the diagram, PQRS is a rectangle and SR = 3 PS. R, Q and Y are collinear points.
XQ = 6 cm and YQ = 8 cm.
i.
Prove  PXS is similar to  QXY.
ii.
Hence find the length of PS. †
« i) Proof ii)
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8
cm »
3
M_BANK\YR12-2U\APPLICATIONSOFGEOMETRICALPROPERTIES.CAT
40)!
2U00-1c
X
53
NOT TO
SCALE
Y
Z
108
41)!

V
U
W
The diagram shows XY parallel to UW,  XYU = 53,  UZV = 108 and  ZVW = . Find
the value of . Give reasons.†
«  = 161 »
2U00-6b
D
C
A
42)!
43)!
X
B
NOT TO
SCALE
Y
In the diagram, ABCD is a parallelogram. X is a point on AB. DX and CB are both produced
to Y.
i.
Copy this diagram onto your answer sheet.
ii.
Prove that  ADX is similar to  CYD.
iii.
Hence find the length of XY given AX = 8 cm, DC = 12 cm and DX = 10 cm.†
« ii) Proof iii) 5 cm »
2U01-3c
A
C
NOT TO
x
R
SCALE
110
B S
In the diagram, AC = BC, RCA and CBS are straight lines,  ABS = 110 and  BCR = x.
Copy the diagram onto your writing sheet. Find the value of x giving reasons.†
« x = 140 »
2U01-10a
A
NOT TO
SCALE
1
D
B
72
x
1
36
36
M x C
In the diagram, ABC is an isosceles triangle where  ABC =  BCA = 72, AB = AC = 1 and
BC = 2x. Angle BCA is bisected by CD and angle BAC is bisected by AM which is also the
perpendicular bisector of BC. Copy the diagram onto your writing sheet.
i.
Show that AD = 2x.
ii.
Show that triangles ABC and CBD are similar.
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iii.
iv.
By using (ii), find the exact value of x.
Hence find the exact value of sin 18.†
« i) ii) Proof iii) x 
44)!
2U02-4c
B
45)!
1  5
1 5
iv)
»
4
4
C
A
D
H
E
NOT TO
SCALE
G
F
ABCDEFGH is regular octagon.
i.
Explain clearly why  ABC is 135.
ii.
Calculate the size of  GAH.
iii.
Using (i), or otherwise, calculate the size of  CGF.
iv.
Hence, calculate the size of  AGC.†
« i) Proof ii) 225 iii) 675 iv) 45 »
2U02-9a
ABCD is a rectangle and AE  BD. AE = 5 cm and DE = 2 cm.
A
B
i.
ii.
iii.
E
D
C
Copy the diagram and prove that triangles AED and BCD are similar.
Hence, show that AD2 = BD·DE.
Find the area of ABCD.†
« i) ii) Proof ii) 725 cm2 »
46)! 2U03-3d
In the diagram, PQRS is a parallelogram. QR is produced to U so that QR = RU.
P
Q
S
i.
ii.
47)!
T
R
NOT TO
SCALE
U
Giving clear reasons, show that the triangles PST and URT are congruent.
Hence, or otherwise, show that T is the midpoint of SR.†
« Proof »
2U04-3a
For what values of a, will ax2 + 5x + a be positive definitive?†
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« a >
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5
»
2