SPCS0271
1)! ¥Yr11-2U\geometry.cat Qn1) 2U84-2i¥
Find the area of the rhombus ABCD given AB=10cm and EB=8cm.
†
«® 96cm2 »
2)! ¥Yr11-2U\geometry.cat Qn2) 2U84-8i¥
In the triangle WXV, YZ=9cm, VX=12cm, WX=8cm and YZ||VX. Prove that DWZY is similar to DWXV and find the length of WZ.†
«® 6cm »
3)! ¥Yr11-2U\geometry.cat Qn3) 2U84-9iv¥
In the figure, AB=AC, BD||FE, BF||DE and ÐCAB=54°. Find the size of ÐFED giving reasons.†
«® 63° »
4)! ¥Yr11-2U\geometry.cat Qn4) 2U85-7i¥
PQRS is a square with PQ=1unit. Find the perimeter of PTRS.†
«® »
5)! ¥Yr11-2U\geometry.cat Qn5) 2U85-7ii¥
ABC and ABD are two triangles, X,Y andZ are points such that XYêêCB and YZêêBD. Prove that XY:YZ=CB:BD.†
«® Proof »
6)! ¥Yr11-2U\geometry.cat Qn6) 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)! ¥Yr11-2U\geometry.cat Qn7) 2U86-5ii¥
A,B,C are collinear points. BD||AE, BA||DE, BC=BD and ÐBCD=58°. Reproduce this diagram on your answer sheet and find the size of ÐDEA.
†
«® 116° »
8)! ¥Yr11-2U\geometry.cat Qn8) 2U86-5iii¥
In the triangle PSU, QR||SU, SP||TR, ST=7·5cm, PQ=10cm, PR=12cm and UT=15cm. Find the length of SQ giving reasons.
«® 20cm »
9)! ¥Yr11-2U\geometry.cat Qn9) 2U86-5iv¥
GL is a median in DHFG and HJ||FK.
a. Draw a neat sketch of this diagram on your answer sheet.
b. Prove, giving reasons, that KL=LJ.
«® Proof »
10)! ¥Yr11-2U\geometry.cat Qn10) 2U87-5i¥
In the diagram AB||CD and GH^AB. If y=25 find the size of ÐGMH. Hence or otherwise find the size of ÐMFD.†
«® ÐGMH=65°, ÐMFD=115° »
11)! ¥Yr11-2U\geometry.cat Qn11) 2U87-5ii¥
PQRS is a trapezium with PQ||SR. Diagonals PR and SQ intersect atT.
a. Reproduce this diagram on your answer sheet.
b. Prove, giving reasons, that DPQT|||DRST.
c. Hence, find PQ, given that SR=36cm, PT=5cm and RT=15cm.†
«® a)Proof b)12cm »
12)! ¥Yr11-2U\geometry.cat Qn12) 2U87-5iii¥
In the diagram below, ÐUXY=ÐUYX and XZ=YZ.
a. Copy this diagram on your answer sheet.
b. Prove that DUVYºDUWX, giving reasons.
c. Hence prove that DVZW is isosceles.†
«® Proof »
13)! ¥Yr11-2U\geometry.cat Qn13) 2U88-6i¥
In the figures FG=10cm, EG=15cm, EF=12·5cm, IJ=4cm, HJ=6cm and HI=5cm.
a. Draw a neat sketch and mark on it all the given information.
b. Show that DEFG|||DHIJ giving reasons.†
«® Proof »
14)! ¥Yr11-2U\geometry.cat Qn14) 2U88-6ii¥
The figure above shows quadrilateral KLMN with diagonals KM and LN intersecting atP.
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 »
15)! ¥Yr11-2U\geometry.cat Qn15) 2U89-3b¥
In the diagram above, KN=NM, KL=LM, ÐKNM=110° and
ÐNKL=45°.
i. Reproduce a neat sketch and mark on it all the given information.
ii. Find the size of ÐMKN and ÐKLM, giving reasons.†
«® i) ii)ÐMKN=35°, ÐKLM=20° »
16)! ¥Yr11-2U\geometry.cat Qn16) 2U89-6c¥
In the diagram above, SQ^PQ, RU^SQ and PS||QR.
i. Prove that DRQU|||DPSQ.
ii. If RU=xunits, QR=yunits and PS is four times the length of RU, find the length of PQ in terms of xandy.†
«® i)Proof ii) »
17)! ¥Yr11-2U\geometry.cat Qn17) 2U90-2c¥
DKLM is an isosceles triangle with KL=LM, ÐLKM=80°, LN bisects ÐKLM and ÐKMN=20°.
i. On your answer sheet, draw a neat sketch of the diagram above, showing all the given information.
ii. Find the size of ÐLMN, giving reasons for your answer.
iii. Find the size of ÐLNM, giving reasons.†
«® i) ii)ÐLMN=60° iii)ÐLNM=110° »
18)! ¥Yr11-2U\geometry.cat Qn18) 2U90-5d¥
PQRS is a quadrilateral with PR=QS, PQPS and SRPS.
i. On your answer sheet, draw a neat sketch and mark on it all the given information.
ii. Prove that DQPS and DRSP are congruent.
iii. Hence prove that PQRS is a parallelogram.†
«® i) ii) iii)Proof »
19)! ¥Yr11-2U\geometry.cat Qn19) 2U91-4c¥
In the diagram given below, DABC is a right angle triangle with ÐBAC=90°, CQ=CR, PB=RB and ÐACB=40°.
i. Copy this diagram onto your answer booklet.
ii. Write down the size of ÐPRQ. (No reasons are required in your solution).†
«® 45° »
20)! ¥Yr11-2U\geometry.cat Qn20) 2U91-7a¥
PQRS is a parallelogram. TQ bisects ÐPQR and VS bisects ÐPSR.
i. Copy this diagram onto your answer booklet.
ii. State why ÐPQR=ÐPSR.
iii. Prove that DPVS and DRTQ are congruent.
iv. Hence find the length of TV if PR=20cm and TR=8cm.†
«® ii)Opposite Ð's in a parallelogram iii)Proof iv)4cm »
21)! ¥Yr11-2U\geometry.cat Qn21) 2U92-5a¥
In the diagram above JKLM is a quadrilateral and LMN is a triangle. JM||LN, JK=KL, JM=ML=MN, ÐKLM=123°, ÐJKL=2q° and ÐJML=q°.
i. Copy this diagram onto your answer sheet.
ii. Show that ÐJML=38° giving reasons.
iii. Determine the size of ÐLNM giving reasons.†
«® ii)Proof iii)38° »
22)! ¥Yr11-2U\geometry.cat Qn22) 2U92-7a¥
In the given diagram PQ||RS. MQ bisects ÐPQR, NR bisects ÐQRS and MQ=NR.
i. Copy this diagram onto your answer sheet and mark on it all the given information.
ii. Explain how you know that ÐMQZ=ÐNRZ.
iii. Prove that DQMZºDRNZ.
iv. Hence prove that the intervals QR and MN bisect each other.†
«® i) ii) iii) iv)Proof »
23)! ¥Yr11-2U\geometry.cat Qn23) 2U93-4a¥
In the diagram CA=AD=DB and ÐEBD=20°. Copy this diagram onto your answer sheet.
i. Show ÐADC=40°, giving reasons.
ii. Hence find the size of ÐCAE, giving reasons.†
«® i)Proof ii)60° »
24)! ¥Yr11-2U\geometry.cat Qn24) 2U93-5c¥
In the diagram CT bisects ÐACB, AE is perpendicular to CT andM is the midpoint of AB. AE produced meets BC at the pointP.
i. Copy this diagram onto your answer sheet and mark in all the given information.
ii. Prove that DACE is congruent to DPCE.
iii. Explain why AE=EP.
iv. Hence prove that EM is parallel to PB.†
«® i) ii)Proof iii)Corresponding sides in congruent D's iv)Proof »
25)! ¥Yr11-2U\geometry.cat Qn25) 2U94-3b¥
In the diagram AE||BD, AC||ED, ÐAED=130° and ÐABC=90°.
i. Copy this diagram onto your answer sheet.
ii. Find the size of ÐBAC giving reasons.†
«® 40° »
26)! ¥Yr11-2U\geometry.cat Qn26) 2U94-7c¥
In the figure triangles ACB and APO are equilateral.
i. Copy this diagram onto your answer sheet and include all the given information.
ii. Explain why ÐBAO=ÐPAC.
iii. Prove DAOBºDAPC.
iv. Hence prove OB=CP.†
«® i) ii)Each angle is equal to 60°-ÐOAC iii) iv)Proof »
27)! ¥Yr11-2U\geometry.cat Qn27) 2U95-1d¥
In the diagram AB||CE, ÐABF=75° and ÐBFE=35°.
Find the size ofq giving reasons.†
«® 40° »
28)! ¥Yr11-2U\geometry.cat Qn28) 2U95-5d¥
The diagram shows a rhombus EFGH. A line EL is drawn throughE so that ÐHEL=2´ÐFEL.
i. Copy the diagram onto your answer page.
ii. ÐFGH=96°, find the size of ÐELF giving reasons.†
«® 106° »
29)! ¥Yr11-2U\geometry.cat Qn29) 2U95-9b¥
In the diagram ABCD is a square. AB is produced toE so that AB=BE and BC is produced toF so that BC=CF.
i. Copy the diagram onto your answer page.
ii. Prove DAEDºDBFA.
iii. Hence prove ÐAED=ÐBFA.†
«® Proof »
30)! ¥Yr11-2U\geometry.cat Qn30) 2U96-1f¥
Find the value ofx.†
«® 115° »
31)! ¥Yr11-2U\geometry.cat Qn31) 2U96-2b¥
In the diagram, AB||CD, AD=CD and ÐBAC=120°. Copy the diagram onto your answer sheet.
i. Explain why ÐACD=60°.
ii. Show that DADC is equilateral, giving reasons.†
«® Proof »
32)! ¥Yr11-2U\geometry.cat Qn32) 2U96-9a¥
DABC is right-angled at Aand AD is drawn perpendicular to BC. AB=15cm and AD=12cm. Copy the given diagram onto your answer sheet.
i. Show that BD=9cm.
ii. Prove that DABC is similar to DDBA.
iii. Hence find the length of AC.†
«® i) ii)Proof iii)20cm »
33)! ¥Yr11-2U\geometry.cat Qn33) 2U97-3b¥
ABC is an equilateral triangle. BC is produced toD so that BC=CD.
i. Copy the diagram onto your answer sheet and mark on it all given information.
ii. Prove that ÐBAD=90°.†
«® i) ii)Proof »
34)! ¥Yr11-2U\geometry.cat Qn34) 2U97-10a¥
ABCD is a rectangle with AB=12cm, AD=9cm and AM is perpendicular to BD.
i. Copy the diagram onto your answer sheet.
ii. Find the length of BD.
iii. Prove that DABM is similar to DDBA.
iv. Hence find the length of BM.†
«® ii)15cm iii)Proof iv)9·6cm »
35)! ¥Yr11-2U\geometry.cat Qn35) 2U98-1e¥
In the diagram ÐPQT=126°andÐQTR=90°. Find the value ofy.†
«® 144° »
36)! ¥Yr11-2U\geometry.cat Qn36) 2U98-5c¥
PN is a diagonal of the rectangle MNOP. Ris the point on PO and ÐPQR=90°.
i. Prove that DPQR is similar to DNMP.
ii. Given MP=5cm, MN=10cm and QR=2cm, find the length of PQ.†
«® i)Proof ii)4cm »
37)! ¥Yr11-2U\geometry.cat Qn37) 2U99-1d¥
In the diagram, PQ||TR, ÐPQR=85°, ÐQPS=120°, ÐPSR=75° and ÐSRT=q°. Copy the diagram onto your answer sheet. Find the value ofq.†
«® 15° »
38)! ¥Yr11-2U\geometry.cat Qn38) 2U99-4d¥
WXYZ is a parallelogram. XP bisects ÐWXY and ZQ bisects ÐWZY. Copy the diagram onto your answer sheet.
i. Explain why ÐWXY=ÐWZY.
ii. Prove DWXP is congruent to DYZQ.
iii. Hence find the length of PQ given WY=20cm and QY=8cm.†
«® i) ii)Proof iii)4cm »
39)! ¥Yr11-2U\geometry.cat Qn39) 2U99-7a¥
In the diagram, PQRS is a rectangle and SR=3PS. R,QandY are collinear points. XQ=6cm and YQ=8cm.
i. Prove DPXS is similar to DQXY.
ii. Hence find the length of PS. †
«® i)Proof ii) »
40)! ¥Yr11-2U\geometry.cat Qn40) 2U00-1c¥
The diagram shows XY parallel to UW, ÐXYU=53°, ÐUZV=108° and ÐZVW=q°. Find the value ofq. Give reasons.†
«® q=161° »
41)! ¥Yr11-2U\geometry.cat Qn41) 2U00-6b¥
In the diagram, ABCD is a parallelogram. Xis a point on AB. DX and CB are both produced toY.
i. Copy this diagram onto your answer sheet.
ii. Prove that DADX is similar to DCYD.
iii. Hence find the length of XY given AX=8cm, DC=12cm and DX=10cm.†
«® ii)Proof iii)5cm »
42)! ¥Yr11-2U\geometry.cat Qn42) 2U01-3c¥
In the diagram, AC=BC, RCAandCBS are straight lines, ÐABS=110° and ÐBCR=x. Copy the diagram onto your writing sheet. Find the value ofx giving reasons.†
«® x=140° »
43)! ¥Yr11-2U\geometry.cat Qn43) 2U01-10a¥
In the diagram, ABCis an isosceles triangle where ÐABC=ÐBCA=72°, AB=AC=1 and BC=2x. AngleBCA is bisected byCD and angleBAC is bisected byAM which is also the perpendicular bisector ofBC. Copy the diagram onto your writing sheet.
i. Show that AD=2x.
ii. Show that triangles ABCandCBD are similar.
iii. By using (ii), find the exact value ofx.
iv. Hence find the exact value of sin 18°.†
«® i)ii)Proof iii) iv) »
44)! 2¥Yr11-2U\geometry.cat Qn44) 2U02-4c¥
ABCDEFGH is regular octagon.
i. Explain clearly why ÐABC is 135°.
ii. Calculate the size of ÐGAH.
iii. Using (i), or otherwise, calculate the size of ÐCGF.
iv. Hence, calculate the size of ÐAGC.†
«® i)Proof ii)22×5° iii)67×5° iv)45° »
45)! 2¥Yr11-2U\geometry.cat Qn45) 2U02-9a¥
ABCD is a rectangle and AE^BD. AE=5cm and DE=2cm.
i. Copy the diagram and prove that triangles AED and BCD are similar.
ii. Hence, show that AD2=BD·DE.
iii. Find the area of ABCD.†
«® i)ii)Proof ii)72×5cm2 »
46)! ¥Yr11-2U\geometry.cat Qn46) 2U03-3d¥
In the diagram, PQRS is a parallelogram. QR is produced to U so that QR=RU.
- Giving clear reasons, show that the trianglesPST and URT are congruent.
- Hence, or otherwise, show thatT is the midpoint of SR.†
«® Proof »
47)! 2U¥Yr11-2U\geometry.cat Qn47) 04-4a¥
In the diagram below, AB=9cm, BC=6cm, AD=8cm, AC=12cm and ÐABC=ÐDAC.
i. Prove ∆ABC|||∆CAD, giving clear reasons.
ii. Hence, find the value of side CD.†
«® i) Proof ii) 16cm »
48)! ¥Yr11-2U\geometry.cat Qn48) 2U05-6b¥
ABCD is a rectangle in which AB=40cm and AD=60cm. M is the midpoint of BCandDP is perpendicular toAM. Draw a neat sketch on your answer sheet. Hence:
i. Prove that triangles ABMandAPD are similar.
ii. Calculate the length ofPD.
iii. Using Pythagoras’ Theorem in triangleAPD show that AP=36cm.
iv. By finding the two areas of the triangles ABMandAPD, prove that the area of the quadrilateral PMCD is 936cm2.†
«® i) Proof ii) PD=48cm iii) iv) Proof »
49)! ¥Yr11-2U\geometry.cat Qn49) 2U06-4b¥
¥
PRandQS are straight lines intersecting at pointA. Also PS=QR, ÐPSA=ÐQRA=80°, ÐPAQ=120° and ÐPQA=x.
i. Copy the diagram into your writing booklet.
ii. Prove that DPSA is congruent to DQRA.
iii. Hence, show that x=30°.†
«® Proof »
[[End Of Qns]]
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