Revision Paper 12: Congruency, Similarity and Circle Properties

Ql ABCD is a square. DE intersectsAC at 0 and angle OEB = 120°.(a) Name a pair of similar triangles, giving reasons.(b) Given that AO = 2 em, CO = 7 em and DE = 7 em, find the length of OE.

(a.)LADE: :: LCOD (vuf. opp. L.s) D C (,b) OE = AO

LO<\E z: L C:cD LeAH. Ls) on CO

LOEA :::LODe (al-\-- Ls) ~ - ~"l-OE I

., ~ME /V .6.CcD CAAA)8[< ., 0 E z: 14-- :20E

90E z: lit

DE - J!±...- Cf CM '9-A~----~~E~----~B

Q2 Two cups are geometrically similar in shape. The height of the smaller cup is 10 em and theheight of the larger cup is 25 em.(a) Find the ratio between the total surface area of the two cups.(b) The smaller cup is filled with syrup and sold at $0.80 per cup. Adam wishes to buy

the larger cup with syrup. How much will it costs him?

(a') ~ = (~);tA] )5

_ it--;)5~-noo? .srn:Jtec o.t·eo.:k bo/a~

C 6) C2. _ \J 2- _ (:Jfi\ 3

CA - V. - IO-J

C~ =-l~~)~ ~O' S-

~$L2'5b*,S 4- ~;l5~ It l;);\1 G:>6+ him -$\)-50 '*

Q3 PQRS is a parallelogram as shown below. Given that BR = 3 SB and SA = !SP .4

(a.') .5 B \ (. \r--_---;;;.-B_~I----~ R SR -=:; tt .5 \\icZf1 )

(b) A.B _ SA

(a) Show that MAB is similar to MPR, p~ - sP. AB= _I )( Cj

(b) GIven that SA = 2 em and PR = 9 em, find the length of AB , . 't-

c. Area of L\ SAB :::: d';;25 LrY)~IJ1'J Find the ratio

Area of parallelogram PQRS ~(c") Area.or ASA13 _ L\ 2 I

Area. cf A .srI< ltt- J = 1["

Area 4- b.SAB .-L-Af~a{ Pa fl-S - .3 :2-!iR

L5 '\6, Lon1h'1otl

, _ ASABrv 4St'{<. (sAs) ~plC:- ~~+- ..../

1

Q4ii,) L ABC:: lWo- b( 0 CcPr L.s ~

o cyeLk 1vOd')s: \\I~ '

B IV) LPAC =- qoo+ :1\0 Clonl.. rod)::::\l \ e

(\) LA~=lt8°(L.sifloJ+. S~) D

LOAB::: 90°- .It-8° (-tan.l red)= ttJo

.. L- VAc::: 2\ b («: biseos at\.B)

=r L OCB z» LtSf>+ .21 0

:=- b,o>j< P

Ii) LCOA::: (~O"_.2I°_)lo(sumofLs in 4")

::: 13g0

In the diagram, 0 is the centre of the circle, PAT is the tangent to the circle atA, AB is

LADe == 6~(j paralleltoOC,LBAT=48°and ACbisectstheangleOAB.

(LoJ Ger'lh-f..Calculate, stating the reasons clearly

::= 2D4 g(e) (i) LOCB,

(ii) LADC,

(iii) LABC,

(iv) LCPA.

A T

Q5. ~~,.

In the diagram, PQ = QS, QR = ST and L PQR = L ~= 90°

(i) Name two congruent triangles and proof the case of congruency.(ii) If LTQS = 37°, find L PRQ.

it) L PRG z: 906_ 310

=: 5 3~(.~um~.is if) b")(i) ~ PQR.;: A~ST (SA»

PO) ~ QS ~I"e() '>QR ~ 6T (~~V~)LP~JZ = L&ST (~Iven)

p

Q R sQ6 Alex bought two cylindrical cans of baked beans from the supermarket. The cans are

/

geometrically similar to each other. The height of the small can is ~ the height of the largehi C\ . 15

can h~ z: 15 C6lven)

(i)

(i) Find the ratio of the radii of the bases of the cylinders,(ii) Ifthe volume ofthe small can is 270cm 3, find the volume of the large can.

R, -.:L _ .3R2 - \S - 5 *V.:z. _ (~\3=r:': 3)V - (Ii:\.3 ;t,O

2 - --s-) ><.

2

31250 em ~

Q7 In the diagram below, MQR is inscribed into MBC such that APQR is a parallelogram.BQ = 6 em and QC = 8 em. . Dr, CQ g 4-

(I) ,l)( - - - -;; =-=1A AO - @ - h' ~

(0.) PR'IQ commcy)

LAPK ::::LQRf> [Ql-t. LS)

LARf> = L Q PR (~Jt,is")

-. ~ P~R. ~ bRAP (ASA") >tx

lb') L R£)Q = L~(5)c.. CWrr. Ls)L PCQ = L 1<-05 C~vT.is)LCpa -==L QRe:> (SUrYlq. is ·'0 A oIf.- neb L.sOh~ . ,:~ or 0.1-\-Ls)

~ R-G.B"-" il i=tQ (MI\)"(a) Show that MQR == MAP.

(b) Stating your reasons clearly, explain why MQB is similar to MCQ.

(c) Find the value of J") AreQ ~ A A'BC = .4t- )<40PQ ~AB' :::: \1-?--5 cw\(i)

, .

Iii) Area~ Ll PQ I<t\f'€Cl ~ ;bRA? - \~6cm

(ii)area of MQCarea of MBC'

(iii) area of MQR ~ - _ '+ 0area of MAP -: ~6... t\? &9-.:::. l '2-?-'-S - :?-? S--

=- CoO~(d) If the area of MQC is 40 crrr', calculate the area of the parallelogram APQR. ~

(e) What is the special name given to the quadrilateral ABQP? T~e.z.l v..YY)

Q8 PQTS is a trapezium with TS parallel to QP. The diagonals PT and QS intersect at R.

(a) Write down a triangle similar to A PRQ and proof the case of similarity

X Give reasons why the 2 triangles are similar.

&3) ~ TRS I'-' AP~Q CMA)L TR.S .:::L fRo.. (~- OW' is)

LJ?,S 1 ::: L RQ P (Ol\-L J...s)Ls\'R :=: L Qf>p.. (oJ\-. J....s)'*

P

Q

3

A, B, C, D and E are points on the circumference of a circle with centre O. The lines AD andBE meet at F. Angle ABE = 42°, angle EFD = 70° and AE is parallel to CD.

C

Q9

e\) LA'DE := Lt}b (16 10 S04'Yl€~j)

LAED;::.'10~lrt. L;I"\~I,J(.

LPAf- :: ~o (sum q.l.s B

\np)

LBfA z: 70<J(vtdr,

°P?'~~L~~~--;o~---;;;Y10

-7 L e,-t\~ ::. (;0°(J VrY] 4UlnA)

~ L AE;3 ::: (toO _ ~g,0_ tt2~ 8°

-:::~do~ (SU'N.. ~ k il'\.&

EStating your reasons clearly, find(i) angle AEB.(ii) angle ABC.

QI0 (a) In the diagram, the points P, Q, Rand S lie on a circle with centre O. The tangents tothe circle at P and Q meet at T. POT = 50° and PRs = 64°.

Find, stating your reasons clearly,

(i) POQ,

(ii) PRQ,

(iii) SPQ,

(iv) SPO,(v) PTQ.

()) OQ:::oPCrod)QT = PT (+0.0 B" ~1 . p-\')