1968-06-06 — Page 31

6 JUN 1968

其三鲜强八第

日一十月五年申戉層夏

WAH KIU YAT PO

CITY: HALL

期量

日六月六年八六九一股公年七十五國民路中

英中會考數學(三)答案

ཚÇཙྪེད།ཀཽ ཎྜ

英中會考數學(五)試題

Į, MATHEMATICE, 'H,

RAPER (

time allowed - Two hours,

·歎陽谿文·

Suzrasted Answers

Mathematics

Paper III

Geometry

B.K. Certificate of Education Examination 1968

Section A (40 marka)

Candidate No.

Attempt ALL questions in Section A and any FOUR questions in Section B.

SECTION A 140 marks)

*

Any

Credit will only be given for the correct answer. For questions 1 -9, put the anaway in the space provided on the question paper,which must be handed in. working may be done on the last few payne of the answer book but will not be marked,

1.

( marke)

Given that ABCD to a syclis quadrilateral with diagonale AC, 80 intereacting.

at P. Pul a "u" in the bas oppoelle se the one of the following statementE

(a)

#b) •

BAC

BPC • 8 2 804 KANUARY*

起手 AP AC

Adi

AP PC . BP. PD

AP PO •

ODDOD

Credit will only be given for the correct answer, For questions I - 9, put the answer in the space provided on the question paper, which must be handed in. Any working may be done on the last few pages of, the answer book but will not be marked.

1. (2 marks)

Given that ABCD is a cyclic quadrilateral with diagonale AC, BD Interesating at P. Put "\" in the box opposite to the one of the following / atatements which must be true,

(a) ( ĐẠO Ở ĐẠO

(b) BPC

W

2/BDC

BD

(c) AP AC - BP

(d) AF PC - BP • PD

3.

(2 marks!

In a fixed plane it is given that A B is a fixed line and that the area of PA B

la constant, Put in the box opposite to the one of the following statements which givɑs the complete locus of P.

اها

the perpendicular bisector of A B

()

a vietuias are with  B as a chard

te)

a straight line parallel to AB

the interior and exterior bisector of CAPE

• pair of lines paratlal to AB

2 marks)

00000

Given that ABCD 14 a parallelogram with diagonşia AC. BD intersecting

at H Put in the box opposite to the one of the following #təlamente which must be true

1/

DAH-

BAH......

[bl

4 BCO • cystic quadrilatora).

fol

area of A ARB

area of A HOD

(4)

AB DC AD BC.

AS AD BH HO

( marka)

00000

in figure 1. ABCD, AEF are straight lines. C is the centre of the circle. AEF touches the circle at E. If Z BAƑ measures a". pat i

in che

boy againal the due of the following supressions which gives the monours of

DEF.

{10}

(c)

jel

none of the expressions {x}, (b), (c), (d)

1.

sé marks)

00000

When Broduced two equal chorda AB, CD of a strele meet at an angle of 20

Has the mid-point of AB, K is the mid-point of CD, calculate the wij? of HKD.

#4 markal

Answer:

LA ABC. M ie the mid-polat of BC, # w » point on AC auch chan AN - ING. If the area of A ABC sa 90 square inches, lnd the area of

4 MASTE

Lis figuri 1. ABC. CDEF are straight lines. Fă biasota az ABE, DBbisscra

„CËÁC, I BË 1 bem, Bɛ a ecm, and EC • dom,, calcuinus tae lengths of ED, 57,

Laasmer

MA ABG, AB • AC. Eu a porni on the alismeta All, - 7 in the zoor et """ the perpendioullar from B to AS U AB420 4.42146, salouisse she ratio

CAM is a median of ▲ ABC. I AB + 9 inches, AM * AG » 1 inches,

salbulate the Jungia of 80.

30, 16 márks)

T

1 is the centre of the inscribed circle at ▲ ABC, AB = 3 fåchat, „c B + 45°, \ucc ́A = 60°, · Construct, with compasses and juler saly, the circumscribed eircis of ▲ 1BC. Flad its radius by makazemans

SECTION 8 (60 masks)

Do any TOUR questions from fala section, Mari auch new question on a new page. All necessary working must be clearly shown. Marks will be deducied for poor presentation of material.

it. (5 marks) |

ABCDEFGH tá à engulas octagon 'inscribed in a circle, centra O. The tanganip me ♬ and E to the strale meer as 22. BC and XD when produced" mesi si Ch... Prove that 80 = 30,

38. NS magics)

LA ABC, H, K are points on the side BC such that BH CK (BH « SKL The circia AMK cuta AB at E, AG ́at, F.

(e) AP • PD - BP *

PC

2

(2 magka

78

(a) is true if ABCD is a square.

(b) is true if P is the centre

of the circle ABCD.

(c) and (e) are special cases, Statement (α) is what we

called the

property of

intersecting chords of a circle.

In a fixed lane 11 is given tha

a fixed line and that the area of A PAB 18 constant.

" in the box opposite to the one of the following statements which gives the complete Locus of P.

(a) the perpendicular bisector of AB ..... (b) a circular arc with AB as a chord .... (e) a straight line parallel to AB

3298 0

(a)

the interior and exterior bisector of

(*) a pain of lines parallel to AB

Since the area. (AB) is fired. a fixed value.

constant

andTM the base the height is then of

the locus of the verke a pair of lines (anc above the offer below the given line AB) which are parallel HOAB:

Pis

(a) is

case for equidistant from A+B case for the vert, a given

(by is the ces is the case for equidistant from slims PA

{marka}'

and PB

"DEF = £2DCE

Cemalt, seg. etc >

< DCE CEA + <#

1 × DEF = ±(90° + x°)

5. (4 marks)

A

when produced two equal chordg AB, CD of a circle meet at an angle of 24. If H is the mid-point of AB, K is the mid-point of CP. calculate the size of

D

AD.

Let be the centre'

AB=CD o is equidistant from AB.CD elies on the bisector of « APC A OPNE A OPK (RHS)

OPHK is isos. <_HKP • £ ( 180°– 24*1

778°

6. (4 marka)

In

ABC, M is the mid-point of BC, N 15 a point on AC such that AN - 2NC, If the area of ABC 19 90 square inches, find the area of MNC..

=454.

Join AM, MN

« AMCŹ of A ABC =

•+ of AMC

AMN C==*

Ang.

15

ag.ing.

1. (6 marks)

- 6 cm.s

In figure 2, ABC, CDEF are straight lines, FB bisects LABE, DB bisecta EBC. If BC BE 4 om. and EC 8 cm., calculate the lengths' of ED. EF.

Lot EDI Cm

then DC = (8-1)tm EF = y to By int« bisector

CF = (8+y) cm

BC BE = CD: DE 4: 4=(8+1)}; X

X=3.2

By ext a bisecta · BC: 8E = CF3 EF

6. 4 =(+8); }

Ane, Eu

EF.

·

3.2

16

Ca

In ABC, AB = ̈AC. E ie ́■"point"on" the altitude ADF is the foot of the perpendicular from E AB. IF AB 1 BC 13 10, calculate the ratio AP ↑ EF.

LIBRARI

Given that ABCD Sa a parallelogram with diagons I' 45 BD intersecting at H. Put a "y

" in the

box opposite to the one of the following

statemenșe which must be true.

(b) ABCD is a cyclio quadri

(0)

area of ▲ AHÐ – area of HCD

(4) AB - DC AD BC

(0) AB AD - EH : HD

K>Y(2`uarks }'!

00000

(a) and (d) are true only

fo ABCD is either a squar

rhombus

0

(b) is trus for ABCD is_a} __Mctangle_or__square_

Q

In figure 1, "ABCD, AEF are straight lines, Cis the centre of the circle, AEF touches the circle at E. If BAF measures x, put a "" in the box against the one of the following expressione which gives the measure of DEF.

(a! 45°

+

2

(b) 45°.

(c) 45° (a) 90°

- x

none of the expressions (a), (b), (c), (a).

AE AD

A

*: A REF In AABD (AA).

1

BD

!

Let AB = /DX

In A ABD'

AB: BC = 13 110

Hen BC#/DX, ..B) #5%. AD = √(3x)` - (57)* =12* Hance, BEAD = **

Ane.

12:5

19. (6

AN is a median of ▲ ABC. If AB =

mátka)

9 inches, AX AC = 7 inches, calculate the length of BC.)

By Apollonius' Theorem

AB*+ AC* = *(AM's BM))

BC=Y 8

Lor by Pythagons' in ri. & ADC, ABD

AD=7'-X1- 9*- 1** g) where y=ax.)

Ans.

Tis the centre of the inscribed circle of ABC,

LA AB 3 inches, L_B - 45°,

60. Construct,

with compasses and ruler only, the circumscribed circle of A IBO. Find its radius by measurement.

10. (6 marke}

8க்

Measurement rad

44 116 markai

Provi, that BEBA CF CA

Hence or otherwida prove than

#EFUBC, then BE GP,

Fig 3

In figure: 3, AE to the bisector of BAC, "PEN/AC,

E POA : 90% Prove inat PB4AQ

RUJ

Two straight lines AB, AC intersect at an angle of 54*. Using the result of part 1), or otherwise, construct a line perpendicular to AC. cutting, A Buat P. AC ar such that AP FAQ = 2,5 inches,

·Measure AQ. Nu description is necessary)

14, 115 market

AÓB 'ia a diamoter of a circle 'centre 0. A chord PQ cuts AO at T such that QTB - 45'.

2008 ** find the angles al ▲ PAQ in terms of a j (it) Hence prove that PQ > AQ > AP,

19. (19 marks).

PB

The diagonala AC, BD of a quadrilateral ABCD mest as" E.

b. 15 marka)

រ

(all

Prove that AABC. A DAC • BE. OT

u ex. EY are altitudes of A BCD and ▲ EDC vespectively,

prove that

quadrilateral ABCD, A DAC •

Prove that, "in"figure 4. 5. K. T. Hare concyclic Wit is also given that PA . QA AB. prove that

(1) 2 HTK 90**

MU, ABAT.

BXEY.