1966-04-19 — Page 15

县“......·日九十月四年六六九一四公年五十五國民骂中

華僑教育

一九六六年度

英中会考

【加大大手區 試題預習專欄

數學科 (FX)

MATHEMATICS (16)

-ESSON 80

歐陽 文

CONSTRUCTION & locus

SINE & COSINE Ruips for a.

In a plane, a curve is said to be the focus of a

moving point if, when the point satisfies a given law, it

lies on the curve, and conversely, it when the point is on

the curve, it satisfies the“ given law.

;

At this stage, we

consider only twà locus-thegrams: cas Perp. bisecta locus (b) bisector lacus. If the position of a point is given by two distinct laws, it may be possible to construct the point by Jaking the intersection of the corresponding loci.

Example 1 › MABC is squilateral. D. E are prints

an AC, AB resp. such that BE=AD. If BB meets CE at P. fing the locus of P

Solution At P be a pt. Satisfies the given law.

Then,

▲ AB ÞÈ 4 BCE (SAS.)

Hence, 8ĘC ZADB

.. E, F, D, A are contyclic,

« BPC - 180 “A = (zoʻ

2. P. lies on a segment on

BC containing an angle of 120",

8

A

skin "If p′ is an arbitrary pt. on the above said

Produce BP' to cut ac at Dt op! Segment.

tuts 98 at E', Then,

<A+LÉP'D*= 60k/20 m180

A.E, P, are caneyelie.

= AB+BC+ CA = HB+BLFCK =10 cm.

9 CACDS 2«K«45°

From Sine laws;

Example 2: Draw a parallelogram vath Sides 4.5 cm.

6 cm one of the angles is of 60°: Construct a rectangle equal in area to the

parallelo- gram and with one side 3 cm. long.

F

N

Given i

#gram ABCD with AD - 6 cas

AB = 4. C Cam

To Constuct · A rect, equar in area with ABCD

and with one side equals 3 cm. Construction • (1) Produce BC hr se CF = 3 cm

-

long.

(3) Join FD and produce it to meet §Â produces

731

at G.

Complete the gram FAGH.

is Construct rect. DMNE on DE, quat

in area with DENK.

Then DMNE is the required rest

Rør (1) rect DMNE=ligram KHED = Ilgram ABCD.

(2) De #3. CAK

Example

Construct a

circle to

•

Freu ̈e, sum ef a

Sin A Sin C

Stu 42°30′

A

Şiir A =

AC 36°21'

4695

A = 180 - CAMCI

=180- € 36°e/' + ** 2o30) = 101°2

Note: (1) Start, whenever possible, by finding the smallest angla

which cannot be obtuse.

12 Calculate the angle opprite to the smaller of the

(3)

given sides as in Example 7. This cannot be obtuse.

As a check, greater side is always opp, to greati. angle and the converse). And as cannot have

twa obtuse as.

Example & the angle of elevation of the top k of a vertical mast from two points A, B on the same level as its foot H, are 32°14′ and 22°28' resp. If the height of the mast is 52 ft. and the bearings of A & B from t are 270° and 220° resp.,

find the distanc AB and the bearing of B from A.

Selection: InaAHK" AH = 52′«Co£ 32°14'

= 82.47*

IM & BHK BH= 52′CH 22°23

And

=/25.7'

A

« AHB=-70 – 220 = 56*. In a ABH, from cosina Law.

£2*= AM* + BH` - 2· AH BH・ Cos A HB

= (8747)+ (1257) – A (84;47) (125·7) Cos 50 - 9272

·· AB *_96.29 ft.

Dom Dine Law:

Sin HAB GH

SIA AME

BH » 200 ANG 1

品

52

Example

<< & E' c = < ApˆË......

• ABE'C 24 AD'S

BEL - AD'

Est, a of eyekt

AAS.

Hence, any pr. on art BPC satishes

law. From t› rải) we conclude that the locus of p is the segment on oc containing an angle

Centre A, at a

iven point

and a secon

P

given circle.

centre B.

are fixed poim

variable Chord of the circle

ucks if necessary

Complete lacus of R

Theo

given circle, PQ

circle APQ Bra

Cas shown in fig.

AB li a fixed chord.

Pa is of a given length.

«ARB= 20 +

180-(2-6)

are emcydić

Const

The locus of R is the whole circle ARBR". In the circle AQPBOP! cas shown in fig bɔ

const

centre

a required

Similarly, Jon

Join NE, produce

8 With Centre Ca

<HAB~ 88°59′

•'. The bearing of &- from A is

€ bearing it. NIJE S

The bearing here means true

HINTS & ANS, TO EX & A The yield for 48 stock

6% stock

Which

адам

circle (which

Investing & shat

sené dividena –97

GALLN | sides opp igual «x)

holding

es also a solutions

Example @ Construct a square, equal in area to

a given reato

"onstructies › In the given rect, if 42>10.

describe a semidirekt (WAH.

CB

reduced musta

semicircle at E

(4) Join 42

Note Us This Construch ou

brud

Since

But

ZHAGIKEAG

ADAE - ADAL W rect, AC

const

R. B. R' are concy clic

The whole circle is agam a locus of R. from (a) & (b) the two circles together repmient He complete locus of R.

Construct a triangle ABC in which <8-30),

We have an al

ahve method for this

Produce AB to % 5.7 42=80 With AX as diameter describe

a. semicircle

CB produced meets the Semicircle

(0) DA BE des

ABEF & He

construction, depends

AK

ABH ACA

are.

of

seating chords

proportion

Example 7 Given

two give.

Solve

rach that

* LAKH

of 45=217-

HK at B

Solutio

the AB

Cost £600

casts £600 fo Shaper shoed at 8:

TAMIL GARIN

Suppose the mon snaight forwa

we have

Exercise 88 Construct an isosceles triang permere

ARCI

height

that it is correct. The radii of two circle distance between their

atre's

and exterior commu

easure their lengths (Compart

length val Compare the

of the

find the locus

angents

mbus

Calculate

iagonals of

Page 15Page 16