1 APR1262
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玄三笫張四路
CITY HALL
日九初月三章来了歷式
▶英文中學會考試題預習專欄
數學科
**
( 廿五 ) ·歐陽鋊文 ̇
́MATHEMATICS (25)
LESSON 25: CONSTRUCT LUN — LOCUS
In a plane, a curve is said to be the locus of a moving point)
if, when the point satisfies a given law, it lies on the curve ;,
and conversely, if when the point is on the curve, it satisfima/ the given law.
At this stage, we consider only two fundamental_1ocus-thecrona: Į (a) Perp.bisector locus; and
(b) bisector locus.
If the position or a point is given by two distinct laws, itk may be possible to construct the points by taking_the_intersection of the corresponding loci,
EXAMPLE 1: ▲ ABC is equilateral. D,E are points on AC, Ab resp
such that BB AD. If BD meeta CS at F, find the locus of P.
with siz k
SOLUTION: (1) let ♬ de a point satisfies "
the given law. Than,
ABD BCE (SAS)
Hence, [BE0=/ ADB s
E,P,D,A are consyelic,
'. [BPC➡180° - £A=120*.
Plies on a segment on BC
'containing an angle of 120*.
(2) If P is an arbitrary point on the above said segment)
Produce BP to cut AC at D'; CP! outs AB at R.
Then, [A + [Z1P1D1 = 60 → 120 × 180°
A,E,P,D' are concyclic
BE CAD B
ABEC ÅÐ1Ð
BE' = AD'
(ext. of cyclis quad.). (AAS)
(corr. sides of #aj
nence, any pt. on arc BPC satisfies the given law, From (1) & (11), we conclude that, the locus of P is the segment) on RC containing an angle of 120°,
EXAMPLE 2: 4 and d are axed points on a given circle; PQ ia a
variable chord of the circle of given length; AP, BQ.. produced if necessary,meet at R.Find the completo locus of R.
R
fig B
SOLUTION; (a) In the circle APCBP1Q'(as shown in fig.a)
**AB a fixed chord
ANS
EXAMPLE
Pe la af given LengtŹ PAG = constant
•*. £ AQB en oc— constant
RxB=constant
L&R'B='+Ź P'AQ'
={180 −➡) +$
#180 - (➡ - §) = qonstant
A,8,8,8 are concyclic (opp./s supp.)
And, the locus of R is the whole circle ARBR'.
(b) In the circle AQPBQ!P! (as shown in fig,b)
ARB + constant
BLAQ B - L· P1AQ'
7180) -8
=180 = (<B) constant
A‚‚Я' are concyclic
(ext, f of a )
And, the whole girdle ARBR1 18 arain part of the plnenis of R.
The two circles (from a & b) together form the counte locus of R.
PLE 3: Construct a triangle ABC in which / B÷30; / C45
perimeter 10 cm.
Constructionil) Draw NK=10cm
Then,
2) from a AHK on HK such that
KI
184. EVERK-SK
之人材,所分別者
---
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螭 曹璐院院長袜子光籍地政府小予粥,
中文中學會考試題預習專欄
1
英文科
(廿五)
、鐘英。
NG
£ ARK = ± of 30—15*, £ AKH=★ of 45 » 224**
3) Draw bisector of AH to meet HK at B.
4K "
ABC is the requirea
Proof both. ABH, ACK ara 1aoa........
parimater➡AB+ BC+CA≈ HB+BC + CA #LO CA and, ABC=22 H-30; AGB=2/ -
EXAMPLE 41 Draw a parallelogram with sides 4.5 cm, 6 cm; one or the angles. is of 60°. Construct a rectangle equal in area to the parallelogram and with one side 3cm long.
[Cavani //gran ABCD with AD = 6 cm
2∙A 60* ......AB 4.5 cm
Ta construct: A rect, equal in ares with ABCD
and with" one side equals 3 de long")
Construction: (1) Produce BC ep 7 8.1. CF = 3:00
Hence)
(2) Join FD mid produse. it to mestsDA prodused at G (3) Complete the //gran FBGH which,
(L) Construct rect.DINE on D5 equals in area witn
DENK
CNB La Che
EXAMPLE:53 Construct direla to touch a given girals (centre A, at a gáven
a: ascond
Mircle.centre
and produce 25.7.
draw: QBQ!//AP
Join PQ to meet cirole a
gain at
1th cent
Bimia
C,P draw a circle (which
(requ
Le «B»againya bole
ced at
for C, P=C,M, C, PM = /C
sircle. (much le the socort
(251
"tajtas "previous exercise,2
proosad,
to try to get
able to persuade,
producing,
d. get connected on the telephone j f. coma to visit..
g resemble,
h. departed,
J. save,
1. accommodate,
་
2. Huck was sitting on the edge of
2 boat and looking ver niserable. Then Toa cape and greeted has. He returned toe, greeting. The sank into silence for a minute. The Huck remarked that they were unlucky for if they had left the tools at the dead tree, they would have got the money. Ton exclaimed that it was like a dream. When Buck heard the word 'dream', he said that he had had horrible dreame all night, with that Spanish devil chasing them all through then,, and he cursed teat Spanish devil. Tom told him not to ourse the Spanish devil but to find him and track the money. Huok said that they should never find him because he would be frightened to death if he saw him,
withdrew to that the
3. a. When we finished our breakfast, Adele and i
the library because Mr. Rechester had directed library should be used as he sanoolroom, where moet of the bookw were looked up behind glass doors with
the i
bookcase walen was set over and contained, exception
of elementary rything that "ould be needed in the we
Taking up ve
vessel with auch darriaulty in both nanga drank to her ladyship's health in a most respect manor and expressed the words as loud as I could in English, which made" the company laugh heartily and wate noise of their laugh almost deafened me.
o. At the sign of the waster to ask me to come to his trenob- side, I walked on the table and as I was in so great a
Sevalled
from house-ɣindow," garden," stable, and cou-yard, and the school were gathered in full assembly, when Mr. Squears, with (umsll bundle of papers in his band, and fire. S. following with. laanir of danes, entered she room and proclaimed silence.
"Let any boy speak a word without leave," said Mr. Squea.. (mildly, "and I'll take the skin off his back,"
This special announcement had the desired arreot, and a [death-like silence immediately prevailed, in the midst of vktoh
Mr. Squeers went on to say,
"Boys, I've been to London, and have returned to my family (and you, as strong and well as ever."
According to half-yearly custom, the boys gave three feable (cheers at this refreshing intelligence. Such cheers! Sighe
tra strength with the chill on.
"I have seen the parents of some boys," continued Squeers, turning over his papers, "and they're wv aled to hear b their sons are getting on, that there's no prospect at all of their going away, which of course is a very pleasant thing to reflect upon, for all parties."
Two or three hande went to two or three eyes when Squeers said this, but the greater part of the young gentlemen having no particular parents to speak of. were wholly uninterested in the thing one way or another.
said
"I have had disappointments to pontend against, 'Squeere, looking very grin; "Bolders rather was two pounds ten
short. Where is Boldər?”
"Here he is, please, sir," replied twenty unnecessarA LY eager voices, Boys are very like men, to be sure,
"Come here, Bolder," said Squeers.
An unhealthy-looking boy, with warts all over his hands, stepped from bie place to the master's desk, and raised his eye! Amploringly to Squeere's face; his own, quite white from the ravid beeting of his heart.
for be was
"Bolder," said Squeers, speaking very slowly, considering, as the saying goes, where to have him, "Bolder, if your father thinks that because---why, what's this, air?"
As Squeare spoke, he caught up the boy's hand by the aleeve of his jacket, and murveyed it with an uppearance of horror and disgust.
"What do you call thie, siz?" denunded the schoolmaster, iministering a out with the cane to speed the
reply.
"I can't help it, indeed, sir,” replied the boy, crying. Phey will comes it's the dirty work, I think, sir--t leust I don't know what it is, sir, but it's not my fault.'
"Bolder," said Squeers, rolling up his sleeves, and noi8- tening the palm of his right hand to get a good grip of the
You are an incurable young scoundrel, and as the last
see what another will do
surprise all the time that I stumbled against a crust and ring did you no good, we must
fell flat on my face but luckily I received as hurt,
ai was heard, call,
A
b say, was started kigames :Le NOW DEILDU
c, having escaped - Air
d. came, getting, saemed, so till.
e. nade, stagger, foll,
f. has written, beginning.
g. take, is told;
h. will introduce.
Exorcise 25 Comprehension
Read the following pastage carefully answer the questiona listed below:
It was Kr. Squeera's ouston to call the boys together, and make a sort of report, after, évery half-yearly visit ot the (oapital, regarding the relations and friends he had seen, the news he had heard, the latters he had brought dova, the “billa. which had been paid, the accounts which had been left unpaid, and so forth. This solemn -prosseding always, took place in the afternoon of the day after his return; perhaps because the boye aquired strength of mind from the suspense of the gorn or possibly because Mr, Squeers himself acquired greater etes- ness and hardness from gertaià warn drinks in which he used to indulke after his early dinner. Be this as it nav, the haya pärei
EXAMPLE &
ords beating it out of you."
With this, and wholly disregarding a pitiful ory for iespy, Mr. Squeers fell upon the boy and ganed his soundly; not leaving off, indeed, until his arm wae tired out,
"There," said Squeers, when he had quite dones "rub: away, as hard as you like, you won't rub that off in a hurry. Ohi you won't hold that noise, won't you? Put him out, Saike," **
1. What did Mr. Squeere usually do when he returned Trop hie).
half-yearly visit to London?
2. May d he choose the afternoon instead of the morning
total hla boys about the visit?
3. According to Mr. Squeers'e report was there any boy who
would leave the school soon?
4. Who were uninterested at Mr. Squeere's report?
5. What was the first disappoint Mr. Squeare geant?
6. How did Mr. Squears quicken Bolder's reply?
7 Waysdid Mrs. Squeers sointen bis dala before ne beat Bolder?
Why was Bolder punished?
9. What wae Kr. Squeers? Do you think he was a kind sant 10 aplain the following↑
acquire intelligence imploringlY
in full assembly
indulge reflect demanded
had the desired effect twenty unnecessarily eager volu
Option: In the given reet.fr AD ABS.
PUBLICYRA
1) Complete ADOB to be a #quare ADKH,
2) With A as a dianster describe «
semicircle ón AH.
3) Produse CB to meet the semicareis
រ at Bi
4) Join AB- ̧”
5) Describe a square on AS,
Then, ABRI ie the muuired square.
to a given rectangle?
HOTE;(1) Fame construction 18 based en fythagorRA »
Since,
HEA ==TL./
* HAG ** OAB → O SAS
în semicircle
But HACEAG sq. (AP
*ADAR =ADAB = 1 rect. AC)
Jaq, AP .rect, AC
(2) We have in alternativa antaum auT:
this construction (as follows) ›
a) Produce AB to X sit, HIS BE
b) With AZ as diameter, describe
a semicircle.
a) CB produced meets the semcarcae at E.
d) fn Ho, describe a aquare" GBEF,
Then, GREP is the required square,
2, Solution
B
+
This construction depends on either the property of inter-? asoting chords in a sircle, or the mean proportional between (two given lines.
HINTS & ANS, TURET (243)
1 Method A!!
Let Then
B
#at.A
Method
Join AD
BDLBC
prevailed grix
thrashing
• 44BD = + of 6ABC = 8 The 1 of a from hom
Similarly, DACD - # of @ A&c=16
A
The L of D from AC - 1675 = 38 cm
C
8
&
P
B
Preek:
R
SACDA ACE
Solution:
A ABC – APGR = { @ APC + Bt Q
* ASR)
= 17点 thì From pha APC, by Pythagoras
CNOTE: 3, 4, o ne byth, Nas) By mid-phra the length of ste line joining the mid-pets of 8 A
* BC #
JACD POL + AI + X QAD PABBI AN+on+qray
XQRD = QPBY
EXERCISE 25
WAC
1) Construct an isosceles triangle of beight 5 em.and nerimstor
18 cm.
2) Draw a //grain ABUD. Construct points. 2,4 on BC, CD such that
AP, AQ divide ABCD into three parts of equal area. Stats your method and prove that it is correct.
3) The radii of two circles are 2 cm, 5 cm, and the custance between their contres is 9 cm. Construct the interior and exterior common tangente. Heasure their lengths, ( Compare the results by calculation)
4) Construct a line of length 41 cm, and measure 15, (compare
the result with the value in ag. ropte tablo)
5) If the base BC of ABG is given in magnitude anu puszczon,
and 1 BAU La given in magnitude","find the lo
centre I of A ABO.ja
6) Given an equilateral triangle ABC, come truct a point P
such that PA is equal to the perpendicular: FM from P tá Givo, reasonë, jemaatika
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