1968-03-26 — Page 15

日八廿月二年甲 戉壓激

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1768

英文中學會考試題預習專欄

歐陽鐇文

數學科 (廿二)

MATHEMATICS (42)

LESSON 27 TRIANGLES I (The 5 centres).

Important theorems

1, The & bisectos of the three sides of a a art concurem

( Circumcentric theoem) The circumcente is equidistant from

the three Vertices.

4. The bisectors of the intes

( In - centre theorem)

The

of i A are Concurrent

in-centre is qquidistant trom.

the three

sides

3.

The bisector of one bisectors

Gure

Fot concurrent.

and two other Ext

* Ex- centre the orem)

Concament. ( Centroia

The ex-contre is squidistant from the three sides 4 The thret medians of a

theasm) The censaid is to of the way along each

median measwed towards the wrfex

5. The three altitudes of a

Centre themɛmi

dre

concurent. (Och

Crea

<a> 7/2 & ushose vertices are the text of the affehudes is called the PEDAL TRIANGLE of the original a > if the a is acute-angled, the ortho Cenke

inside the ad the ortho cenke lies outside the a for on obfuss -angled ; and lies te vertex of the it for 2

#f.

•<> Orthoce the is NOT the cente of any impatent

circle associate with the a The word

NOTES:

cente comman point

is often used to denate the of intersection of three or mar concurrent lines.

17: If the a

,

in centre each other

the is epimilateral, then,

DLP CAM Centre orthocentre and centoid att coincide with

1: Conversely, if any two of the circum centre, in centre,

arthacentre and centroid coincide and another the triangle is equilateral..

EXAMPLES

Example 1: Prove that the distance of the avitacentre of a triangle from a vertex is turice the distance of the circum centre from the opposite side.

PROOF: Let o, H be the cir cum centɛ.

and orthocerte of a ABC resp. and let M, N ba the vid -plu.

"Jain MN, PQ

In a ABC

Gives

B

* OG «QA-

BP = PC. POLZAL Mid-pt. betrza.

MNAC

dertaily, in 4 HBC,

POMN

*. OQ &#8

+ CE LAB

* OB // CE

~ OPI AD

OP & BC → AD LOC

And,

A

**=2÷WEEK (R)

Z-VASCRI

ES MAR 1968

CITY HALL

日六廿月三年八六九一年七十五國民中

校工亞利多維

日放開行舉昨

二期星

WAH KIU YAT PO

報

實檢,工。在余工, 操以出外,非哨放各

設夲展鏡會、列各項

放日中,除在

,招待各界人士及唾生家 難炁淡的運行硭大放

**2 (2) 熙、宮市等。韃開數席,餓安嗇,就

银 校際問答比賽

進入第二循環

今晚由奔心禳院對聖士

馬、泰六安、盈照騎、王淑嫻、蘇門英、 學術器、市些茲、邵紗旒、邵立涵、遒月

焼開少婦,及有關人上作陪。它方到有

REFRENZ-TAXRESE 湖潮、好近於漢M-B-R、神秘,昨假 致狱灭,以保良局前任主席的朝校校監方 (PA)***KREKEZON

讌賀方潤華獲勳銜

三 方樹泉學校教職員

唯一之雖得者,殊爲光紫云。《谜》 ,做殼予一燦榮黃愷,胡若爲本泄灣生中 星】二月在倫敦時報予以公佈。此項獎金 玆,雖得波爾特獎學金,並由該校於本年 學政治經濟,今夏阿將塌祟。因其用將車 綴均極優異。現在牛津大學三年級激茧 無題,在蘇小港及華仁中午就風時,或 EBEZPERKEHNKEY-0 〔桲匯】本祜族出湖南鄉,篋大律師

胡耀蘇在牛津大學

獲一等榮譽生獎金

胡鴻烈護員長公子 ,下午四時結束關放Ra() SVEDKEERDEEFTOYTOR 黃余,川流不息,由

界人士,凡出 前往該校叄烈的有工

的舞学·工婆品。 EL. #SCESERE -EXTESFREE

提反中學

[353-615

| · FISKERIFIEK

紫嫣離中检查囧筈比拼異常激烈, (Z)-4E÷E-FE

,等

BONE EX

RIB•ZGZAK E

| 卤期心鹌隃型士捷瑪利亞,挽在拉希•

4:46-80CE(24) -HE

(**)-***

比賽的防行头取得衰,何怡關,馮宇風

**

10163 英文中學會考試題預習專欄 1968

英文科(廿二) • MRS · BENNETT •

HONGS

the sides of » ̧OPQ, HMen are il cach other.

the inf

Le.

Ste

correspondingly equal.

PQ = MN

• OPR= 4 UMN

OP = HM{* £ BA] HA = 8.0P

Similarly, HC = 2·00

pravcat

ASA

Or the problem may also be praxed as follows:

With centr 0. radius OB, doaus the circumcircle of ABC Produca BO to meet the circumcinde at x Then, (i) AHCA is 14 Ilgram s.1. CX = AH.

diz a BCR is de at C

OP =÷CX.

and

English (22)

Answers to Paper 21

1. Mr. Richis asked Tom what right Marnot had to be there. Tom told him that Marnot was a big wheel. They couldn't make a fuss. Mr. Richie suspected. that Marnot was up to something. That man was always up to something- Approaching them, Follard, the minister, smilingly said to Monsieur Bichford that he congratulated him. He pumped Bickford's band and excalimed that it was

a magnificent job and was done on time. Bickford said that it was ahead of time, but he trusted that he was to be paid promptly.

2. (1) interrogate (11) necessitate (iv) accommodate (v) intimidazo

(111) fabricat (vi) rotate

3. Ms. Barcastle: So, so, they're gone off, Let

them go. I care not.

Hai Joastle: Who's gone?

Mrs. Hardcastle: My dutiful niece and her gentleman, Mr. Hastings from town-he, who came down with our modestitor here.

4. (1) I received ten foliara in excess of my salary,

No ofte bad access to the private Library.

(11) We repaired a large sum of money for our '17

Cho drive from that benevolent gentleman. fresh air is beneficial to the body The order given is very explicit, so no further explanation is necessary.

Though he did not actually say so, it is implicit in his speech that we should work harder.

iv) Our manager naa agreed to give us an allowance,

to cover the increased cost of living.

It is not easy to earn a livelihood in this place, unless you have high qualifications. (v) It is oply effeminate on that cry,

The feminine gender of the word is unknownIL. (wil The inheritance which he received from his

father afforded him a life of ease.

Be came from a family of soldiers, so his love for adventure was a heritage from his ancestors..

5. (1) under (11) of ̃(141) before (iv) trom (V) **)

(vi) in

Paper 22

The

truth

is

that

the jungle is

neutral.

SHAKHEPEHTE

It provides

any amount of fresh water, and unlimited cover for friend as well as foe - an armed neutrality, if you { like, but neutrality nevertheless. It is the

attitude of mind that determines whether you go under or survive. There is nothing, either good or bad, but thinking makes it so. The jungle itself la neutral.

The first night found us beside the Sungei Sempan. We camped on a sandbank several feet above, the water-line, as it was the only more or less level place we could find. When we bathed in the river, we found many bloated leeches stuck_to_wesiai

-

parts of our bodies. The theɖfy is that" leschea should not be pulled off, as their teeth stay in and】 feater; they should be removed by touching them with salt, tobacco, a solution of areca nut, or a cigarette and. My experience is that the wounds bleed just as such and are just as likely to go septio however_they, are removed, or

I was to learn that navigation in thick mountaià ous jungle is the most difficult in the world - and i had always rather fancied myself at mp reading and finding my way in all types of country from Greenland to Australia, in the first place, it is } quite impossible to find out where you are on the j mp: the limit of our visibility ia fifty to a hundred yards, and even if you are on samm steep hillside, where a small landslide has opened up a window through which you can catch a glimpse of another steep blue tree-clad hillside, you are none the wiser, as one hill is exactly like another. Ther are no landmarks - and if there were, you could not see them. Another difficulty is that there is no WAY of judging distance: it took us more than a weak to realize we were taking sight hours to travel one mils on the cap instead of the three or four miles we imagined, judging by the amount of energy we were expanding. Perhaps the greatest impediment to navigation is that, having decided to move in a certain direction, you are quite unable to do so owing to the difficulties of the terrain: we were

Fontually forced off our course by swampe, thickota,

"precipions, and rivers. It was impossible even to

follow a ridge unless it was very steep and clearly marked. With auch limited visibility it was seldom clear which was the main ridge and we soon found Ourselves down in the valley bottom, baving inadvertently followed a subsidiary spur. In the end 7 found it best to follow the line of least resistance as long as we worked steadily westward; but we very goon had absolutely no idea where we were on the map.

Saumarise, an less than YU words, the difficulties” ör navigation in thick mountainous foresta, Why did the average N.C.0. last longer in the Jungle than the private soldier?

3. What does the writer tell us about leeches? Signy Explain the meaning of the following sentence:

I had always rather fancied myself at mp reading. Explain the following figures of speech used in the passage:

BRA

My experience is that the life of the writish private soldier accidentally left behind in the Kalayan jungle was only a few months, while the average N.C.0. being more intelligant, sight last a year or even longer. To them the jungle seemed predominantly hostils, being full of man-eating tagers, #deadly fevers, venomous anakes and scorpiona, «nativee with poisoned darta, and a host of half-

iamgined nameless terrors. They were unable to adapt themesltes to a new way of life and a diet of riça” and vegetables; in this green hell they expected to dead within a few weeks - and as a rule they were. The other school of thought, that the jungle teem with wild animals, fowls, and fish which are simply there for the taking, and thai luscious tropical fruits drop from the trees, to equally misleading.

nameless LEITOID

green hall

school of thought

a small landslide has opened up a window } follow the line of least resistance

6. Explain the meaning of the following words as

used in the passage:

septic, terrain, impediment, inadvertantly,

1.4.

Example 2. If AD. BE. CF am medians of a ABC, prove",

AD + BE + CF >/«^8 +8C+CA)

Hat

PRODE. Let G be the centroid of o ABC.

From GBC

GB + GC > BC

** 28+ CF >80

Similarly, by sum of hoo sides of a a,

La a GAC

JA GAB7

Adding.

CF + AD > CA

ĴAD+ †BE > AB

(AD+BE+ CF ) > AB + BC + CA

AD+BE+ CF > Ž(AB+BC÷CA)

Example 3. If H. G.O are resp the orthocenke, centroid

ond circum centre of A9BC.

<R> N G. O are

<b> HG - 2.GO

Collineas

"RODE' Let AD, CR be two attitudes

M. N be mid-pt. of BC, AB ng

GA, GA. E. F

In a GHA, by mid-pt. theorm·

EF 4 MA

B

from Example 1, we have HAB & OM

EF BOM

FE O

4 OMG

Prove that

A

And

NOTE

„OG = GF (=2&H) Cor. sides of # A‚' ** SHE 2 GO

OGH is called the "Euter Line" of SABC

Example 4 x y z are the mid-pt. of the sides of ABC; prove that the orthocense of onre is the circum centre of a ABC 'froof, ket o be the orthocentre of axre

Join xo and produce it to Cuf

YZ at P

KPLYŽ

1. Z are mid· pt. of AC BC Ye WAB

( mid - point theorem) XP LYZ and

YZ WAS

PX 1 AB

B

Since I is the mid-point of AB

Px biscots AB at of as

Ox

:

1.2.

Similarly, OY

4

AC

A

(Given)

Hence, is also the circumcentre of a ABC.

HINTS & ANS, To Ex. 2:

MD

1; het n be the mid-pt. of AC

Join MN, ND

•

aft.c.OMIF"

R$

*son, as of = a.

e c » GA = GM

j. « Eƒ 6 SAMOG

LAM # a st. line

and EGF = «M«O

coffinear

Go are

at is the mid-pt. of hyp, AC of danco

2.ND=ŹAC

B

M D

M. N are mid-pt. of CB. ¿A rasp.

(mid• pt. theorem)

2: MN IF AB

-NMD - AB, but INDOMACE «NMD+«MND.

MD=DN=AC

SANAL D = 1 PAND

2,

B

"RD=RA

· OD=QA 2.

Adding - RD Q

RPA MARA:

•QDA <<ÔÁ]

- BAC

NOTE: Subtract, if ■& or «C is

obtuse.

3. Produce CT to meet AB at K, <^B>84)

"/>

AAKTE AACT (ASA)

· KT = TC

KT-TC and BD=DC

DTH BA ? And DT + BK

th

B

- (BA-AKJ={{BA+A<> /

A

PROOF: Produce At vo meat BC at M,

"

AK "

*

14 M.!

4. BAH = ABMH (ASA),

AH = HM

Similarly CAK=¬CNK

1

EXERCISE_22

2.AK = KN.

1. k anal

mid- pt. HKEMN 13. HRY BC)

1, 4O.D. E. F are resp the in-ceme, and ex-censes of a ABC, then © is also the ortocentre of aDEF. 2, ABDE, ACFG are squares on AB. AC outside anac Prove that AH, BF and CD ase, concurrent.

Let AHL BC

3. A. B. C. D are four points on the circumference of a circle Prove that the perpendicular bisectors of me. sc. ni

BC, BD. Cu am concurrent.

4, Q is 4 point inside the paratistogram ABCD such the

<QBC and « GDC are thus prove that AQL BD.

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