1967-04-04 — Page 17

賞一第張五常

WAH KIU YAT PO

fa

育教僑華

一葉中文中學會考試題預習專欄

英文科 (廿三)

·英。

TYRSKY

英文中學會考試題預習專欄

***

數學科 (十三) ·歐陽鐇文·

MAT:IRMATICS (23)

LESSUN 23 CIRCLASS

EXAMPLAS 1: ( en cheras er a elwale)

X, I are the mid-points of the cherds

AB, CD of a circle, centre O; IN, YM

are the perpendiculars fram X, I to

CD, AB resp.,If X cuts at F,

prove that OP and XT bisect sach other.

PROOF: Join OX, OY.

Y is the mid-point of the sherd ABN

OLING

* OXLAB.

- 1 AB (given)

0X//IM

Similarly. beth OF, IN are perp. Le CD.T: OY//XF.

· OX // FT & ÖY {/ IP-

By definition, PYOX is therefëre a parallelogram. Hence, OP and XY bissct each other,

EXAMPLE 2: angle properties of a circle)]

The bisectors of the angles ABC,

ACB of ABC intersect at I and

cut AO, AB at Y, 2 resp.; the circles BIZ, CIY meet again at E. Prove that IXZ+/BIC=2 rt./s. ROOF: With the netāțians as shown..

Join II.

/TXY = c. is in same deg. } ZIXZ = b.

Since, .-,and b, b2 (given)

[YXZ = [XXY → [IXZ - “.+ b,

From ABC, by / sum of a

BIC + b2+ c2-2 rt.₤s

· / BIC + /1X2 = ? rt./s

SXAMPLE 31 ( concyclic points)

ABCD is a parallelogram.

O is a pt. inside ABCD auch that,

LAGB + LGOD = 2 rt.Lo

Prove that 40BG – 4 0DC.

2007: Draw AP DO, BP # CO.

Since the three sides of AAB?

are parallel to the sides of AÇDO

• ABP ÷ + D60

But, ABCD

AABP RADCO

LAPB #DOC

app. #idesgran.

ASA

corr. fa oỈ KHÔNG

LAPB+LAOB/DOC + ₤108 = 2 rt./#

A, P, B, O are concyclic

OC PP constr. & preved OP # BC α= b

Join OP,

is is same segment

-proved db. 1,9. /OBC = 1000.

in whieb, and, & memb

werk24% (chords & arss)"

ABG inscribed in ́s circle as enéwn, Dis the mid-point of minor are BC. O is a point en DA such that DO DG. Prove that 0 is the incentre of aABC. FROOF: Join 00.

miner ard EDwainer are DC (givan)

In the eare circle, equal ares subtend

equal angles (at y

1.5. OA is the bisectar of ¿BAG.

It is required to prove that OC is the bisector of LC.

In AOAG,

In

OCD,

ext./ of a

base 8, is00.

But ¢ ̧—à ̧~8, (is in saus segment)

** 0,= ©2

1.0, oc la the bisector of [BCA. Fram (ə) ado (b), we conclude that,

Dis the in-centre of ABV.

EXAMPLE

ABC 13 an equilateral triangle inscribed in a circle. P is any point on minor arc

BC. Prove that: PA~ FB + PC

400F: Produce BF to D,such that PD = PG

In ACPD, PC= PD

constr.

[CPD-BAC60 ext.[of cyclic PCD is equilateral. quad.

Consider 4APC, BCD:

LACP BCD (= 60 +, ∞ ).

AC=BC

GP CD

A ACP

BCD

(DC-DC)..

-(b)

↑ KALEVSTA "To me ̃previous"exeróisa;

}*] B00GUGS TO: GOUIN BOT.Ko: to, plas]like the others. He had to,

| WAL TOWARD Ina TAROP +

DATA

rance that he was going;to:vbitowäed-made his heart:filled)

with despair

to.1rno immense unwhite-vasbed:fence made bio it on the gox

• di scouTALSU.

*A get'water from"the"town"pump›

Doma |

.. Bo. be used to

I.

DO. 31 was not

Far

Away. It"vs"only"a" hundred "and" fifty yard.

away. But i did not retum, free the Dund at once. Na wang

to fool

around with the boys st the puan.. That was why be.

had to take as nour to. do the NOTAI

I. Too aaked Jim to leX DID O prat STom: the: pung}

↓

ne was tempted by Tom's marviT.

To, because he ese stopped by Aunt Polly.

1. BennUL DA ADA ВАЛ СОда in the right time.toletop: Top true,

running away from his work_

BOO

k. The thought that the free boye would

worm mini 020.2158 150.ĺ

J

ПЕТІВК FOE

i. Dals we having no dope.

dipped

Тормов

innsreed.

the very;top

opera 1 On A g

tririing

CRIAD

immen 81ty

WORK

emaxxi

Long 1 no.

TOO1232 abou

- great amoun

BIỆT - truk

VOITO-a §b

energetica V

umphant

----} Joustne y w;

二期

日四月四年七六九一屉公年六十五——

En room," or even coletanos-elastinTAAN malenTANA KAN (well quite worthily." We can saamaikatġur|Close the genuine bearçon va rewa»»« đo yords tali us.) There¡ie us suku.rineco, no restraint, on either side. The thread of our intimacy bas been snapped. {The leave-taking is an ideal (ma.my co7, (then leave the leave-taking at that? Always,įdoparting friends Implora us not to bother to comA the railway station nexe] morning." Always, we zre deaf to theee entraatioë, knowing thea] to be not quite sincere. The departing friends would think is very odd of us Yo took them at their word.] Besides, they, really] do want to see us again. and thas wish is heartily reciprocated;

to

we duly turn up. And then, oh then,' chat a gulf! yawna) Vel

*

stretch our arms vainly across it. ve hava utterly lost touch

We have nothing at all to say. Ve gaza s1 each other as animale gaze at bunad beinga. Vej'cake conversation'----} end eüch

-Jend

conversation! We know that these are the friends from whom we

parted overnight. They know that we have not altered. Yet,|| the surface, everything is different, and the tension is such

that we only long for the guard to blow his whistle, and put. Any

end to the farsa..

·Hintay

feat

lamentably

in ratio to

1

restraint -

DL:49 flange WW3

entreaties

11.

reciprocatad

f

farcel

MATH

2.Translate the following incathellahi

7

vinh thể önen kandi

tq # #

cover witda mixture?

full or energy, mocessful.

Exerc186 23

Trungista on

1. Translate the following into Chineser

2 GLID po's good as zvito go 11 wall seemo"to"më ons on

the most darriault zaings in the world.and probably seems sol

to you too.

To see a friend.

enougn. Bus ve 379 nov

It is only when a trio

wari

De Bosent for a 12

EDAIWAN Ętations The Gearez,

Frailvar atation were easy

? to perfori tbit anali feat.

ob.aux.ongien Journey O

that we turn up at the

lema and the zonama %D8

likely sokeace, the sarller do we tum up, and the core lament¬

Die ad ve rail. Our failure ta in orast Patia to the terious;

to the depth of our feeling.

nose of tas occasion,

1

静らみ利に好

性嗜酒「家貧不能常得親會知其以此置

ORK HAMMMEL TAN

ABNIKIO

BRARK

generally mosh;am

(nonour "and"riches:

surrounding walls

Į DASİFƏTƏ ̧ANË, KOUSENĮ Corsźre-save(iyatr

8

AP BD,

$ino, 80=BE+ PD =BP÷+ PC

sides of squí. a PCD.

SAS

HOTE; We may také s polat E on AP such that PE=FB.Hence, prove that, BPC AABE LA BPB de equilateral ). then PC AE.

EXAMPLE 6: (tangents)

As shown in fig. 1,8 4re

the centres of two circles

touching the parallel lines

GQH & KPF; the common

tangents PQ touches each

circle as shown. Prove that

(a) AFBQ 16 a rectangle;

(b) PQA8.

$ROOP: Since PK,PQ are tangenta from P to circle A.

AP bisects EPQ.

Similarly. BQ bisecte PQ8.

But

EPQ/PQH

**paq ̈·

AP/78Q

AQ//BP

& AQ//BP.

alt./g.CH//BP)

alt-10 p=9.

Q

Similarly. AQ,BP are the bisectors of the alt.La GQP,QPF,

APBQ is, then, a //gram.

44, 80 ere bisectors of two adj./s on at. Line CQH.

And,

ZACB of 2 rt./a 1 rt./

•APPO de therefore a rectangle, AR=PQ (diaz, of a rect.)

SEAMPLE 7+ (alternate segment)

"ABCD 18 a minor are of a cirozė sucoj

that AB BC

when AB and DC anst "produced, at P, and DB is produced -to pest the tangent if at T.

Prove that TP-TA:

AB BG

stand on equal chorda era equa.

d,- da

At Le a tangent et å

is in alt. sagmen

PROOF

..

A. T, P, D are concyclic

¿TPA de

{ TPA (4,)'

XAMPER 81 (contact of circles)

Vin ARC, AB=p, AC=q, {BAC = 90° and pg:0 is the mid-polat of BC. Gizeleaseze drawn with AB &

AC as diameters.

Prove that two circles can powe drawn with as centre to touch each of these circfes, and,

"/a Lo same segment?

eides oppɔ, equal je

find their radii in terms of p & ą. PROOF: Let E, F be the mid-points of 48, au resp.

Join OR and produce it to cut OE at H, K. Join OF and produce it to cut OF at K, N.

NOTE

OH=QE+J

where OE»‡AG (old-point theorem)

EH-EA

AB

OR( AB+ AC) - -

Similarly, O=OF+ FM

whers OP † AB and FM÷FA AC

OM (AB+ AC)

from (a) and (b), we have OH÷Œ,

Hence, with centre 0, radius OH −£{p+q} We can one such circle,

Secondly," consider now, 08 and 0X.

-

OK EK-XO -- (AB AC) = ±(p −9} ON=0F-HF =÷(AB AC) (p) Hease, with centre 0, radius of fipin), HE ĐI (another circle to touch each of the given öirales,

FO

As shown in figure:

|(9)-

(1) ABC. Is said to be giroumscribed

about e 0.

:

(11) Quad, LMFP is said to be inseribed

in 0.0.

(111) ia dala to be th

circumscribed about IMEP) or, inscribed in a ABC.

0. D ́are în-cushe " ex-owne of DABC

ADDIE MET. 16"

DOA, FAE are the int, 4

angh - brector of 44.

DOA FAE

20h is an

ADEF

Similarly § EDB, rocTM

OF LA DEF.

T'a cu mest, at a

the artho conte DIES

ABDC = ABK}(SAS)

db,

A CDLBK.

Similarly, BFLECA

• KH BF CD an athhudes of akse. 12. AH, BF, CD are con Current

Cat the orto centre of a KBC),

Sa Ket the perp biztatus of 16, BC, AC

meet at o ( sircumcentre of a BBC)' 14 M AÐ, BY wet at P AC, AD, CD MEET NË Q

Then, Pro both lie en de purp bisector of AB, Q TO LOA # on Mo

perp. bisector of AC etc.

P. & bor, cäincuda with O.

NOTE. This pros. in very easy by applying the properties)

of chords in a carele.

The perp bisects of a cand passes through

the

DEMO.

Hence, all ML Con current at the cente

centre of the circle bisi ctos

AZLHUISE 23

are

c) Proof: Join BD.

OD ADC QB 16C 'Hence, Q is the

SABD

DQ produced & R

Bd produced & BD.

orthocentre of

AR 1 BD

a) A triangle ABC is inscribed in a cirole, and the bisectors.

of the Za meet the

at X, Y, Z. Show that the Is of XXL are resp., 90-£; 90 ; 90-£.

(2) Two circles meet at A. and B as show, 40, AD are the diameter,

of each circle. Prove that:C, B, D are collinear.

Two circles meet at A, B. CD is a common tangent to LWO circles. Prove that CAD / CBD-180.-

ABC is inscribed in a circle. O is the orthocentre of ARBU, The thres altitudes AD, BE, CF are produced to meet the circle again at G, H, X resp. Prove that O is also the In-centre of GHK

•

(5) Two equal chords AB, CD meet, when produced, at u.

Prove that BGDG

ABCD is a cyclic quad. If M, N, P, Q are the centres or rom circles inscribed in As ABC, ABD, ACD, BCD. Prove that NFQ [\is a rectangle.Me

The diagonals of cyclic quad, ABCD out each other, at re.fs at P. Prove that the prependicular from P to BC. bisects, when produced, ¿D. (Brachmegupts Theores.)

#) AOB, COD are two prependicular diameters of a circle, Tw chorde CP, Ca out AB at H, I, Prove that H, K, Q, P.... ars coneyclic.

Figure for questions A