1969-04-01 — Page 23

育教僑質三第張六第,日五十月二年酉己曆夏

WAH KIU YAT PO 郭日僑罩

文中學會考試題預習

數學科 (廿二) 歐陽文

MATHEMATICS (22)

LESSON 22 AREAS

IMPORTANT THEOREMA

Arsa theorems on triangle:

= < base x height>

Taron's formula

A, on the same base (or equal basesy and between

the same parallels are equal in area.

Area themems on quadrilateral.

Area of rect.

Length x breadth

Area of syuare = sguan. on one dio

Area of Kgram = base x height

ligram (ine. rect, and sp.) on the same base for equal

bases) and between the same parallels are spust in area

venage of two bases x height

= median x height

Area of trapezium =

Area of rhombus =

of the product of two ding

期星

日一月四年九六九一整公年八十五國民華中

EXAM

km bisects the

TRY IN ARBC, AS=5" BCES

« ABC=45° Calculate the area of JABC and length of AC. (h) ABCD is a guadrilateral in.

which eh=3=90° AB=6" BC= 10* AD=18" Prove that BD bisects & ADC".

SOLUTION: (0) Draw AD LBC

"ADB = 90%

Const.

bau cx} and blew ZAlfCx

A CX2 =d. CRA

of which, ACXY, 7*

By Subtraction

Common "to both

A AXY.

they are on the same pase (ZT) and between the some parallits (?¢ // ZY)

AAXY and A CZY=)

LAXY

Hence

<BAD

BD = AD

sides opp

+(AD)(BC).

HINTS & Ans. To Ex 21

Produce AM to E such that Mỹ anm ABÉC is then a ligram

AB > BE

Hence

x 10.6 (38.m) 丹吉

cas

DC= BC » BD = 6-

rom a EBCT BE +KACSBC From From Adding

CF+ GBA = CA.

AD +ICB > AG

Bythagoras,

ACTM= AÐ" + DC"

=(-5)+(6-=

18.57 HC=W18:57

vihagoras" thesem:

the area of the square on hypotenuse equals the sun of areas of squares on two adjacent sides

EXAMPLE 1 Find the area of a triangle whose sides are (p"-7*)\

*pg, (P+8") inches Grue Prasoo$.

Let a, b, c be the sides of the o such that

ADISC

Take a point x on AD such

JD E BC

XBCD & a #gram:

From H. △ABX

·⋅AX=18-18

AB=6

By Pythagoras.

the

mangled

aea = fb =(pg2p*8*

This problem can also be solved by using

ama formula.

Ascs-ases-b>cs-c)

where

AD+BE+OF Y÷CAB+BC + CA)

<by Produce AD to Ï such What DPAD

Join BP

From AABP:

Ssimilarly

AB+AC > 2 AD

BC+ BAD & BE

CATCBY 2 OF

Adding, AB+8C+ CA> AD+BET+ CF

Hence, in iguam XBCD • BY=8C=/0 (m.) XBCD is a rhombus BD=ding.

BD bisects LADC.

EXAMPLE'S;As shown, ABCD

is a ligram and

Prove that

PR HAÐ RSH AB

(a) & SPA+/65PQ=&SAD APRBA

ASCR=2. ABKD

(C) APER

门

PROOF.

(a) Join KD. PR

ASPD

= AKPD+AMPS +AKSD

às on the same base and beksaan.

the same ils are egust in area.

APR A

AKPS 13 'n

common.

AKPDT AKSD+AKPS=AAPS + 3K5Q+aKPS

ASPA + ASPA➡ SSAD

13. Join KA

A ASQ == AKSA+« KSQ ̈ + 4QKA

OKSA=AKSP #4 of from KSBP oksa = ± of Higrom KSCQ

*4 QKA = 4 QKR, mL of thram MD RK AASQ +▲APR

tal KS BP + KSCQ +ØDRK♣ APKR)

5 of ABCD

ABD

SHPX = DAPY (AAS)

PNBPNC (SAS)

APXBS APYC (RHS)

And

PX = AB - BX = AB-C

= AB-1AY-acy

= AB - AX FAC AX== CASTAC)

and excentre of △ABC

AOD 13 A st. line. DOA, FAE are the ut. ext. anglew blsectors of ch

DOA & FRE

an altitude of Að

Similarly, FOB, FOC are altitudes of ADEF

NOTS

AD, GE, SF intersect, at

the ortop centre

ADEF

AC

AB,AD, BD

1 bisectors of AB,

ACAD CD bisecta of AB

A4

both coïncede, with

In a circle

the I bisacker of

•passes through the cante, S. bisectors are concurrent af of the given circle )

chord all the

Prent Produce HA to K

Such that Ak=BC

4 BDC #4 ABK (SAS)

d=by

7. CDL BK Similarly BFIKO

KH, BF CD am alt. of a KBC AH, BF CD are concurre cat the artho contre of in KBC)

EXAMPLE 21 HD is an attitude of LABC; P Q are point

on AD produced such that DP DA = AC. Prere that BQ =cp

PROCE· IT AB>AC, then from the fig. as shown

ADPL BD C

480, 800, PDC, CIÀ GH IT. AS

by Pythagoras' theorem:

•From W. A BDA: BD" — BATM- AD"

From H4 6RDI BA?

BQ* - BA - A2,

From: vt a CDA

From it. acP DE

C.P2 = ACT

Fring (1) & cay.

DPZAB & AC=D&

CP*DQ AD + AB

線

CA

BQ

By adding the case for ABC AC

then completed

EXAMPLE: 3.

the proof.

shown in figure, ABP is a st.

and M is the mid- point of AP Prove that KM bisects the area of AABC

PRODE: Join PK

A, PMK, HMK ONE OF

equal bases (AMxMA)

and of the same alfitude

AKMP = AKMA

AKMPS AKMBT AKBP..

AŞ KBP, KBC am on the same bate(KB) and between the same /S (BK#PE)

KBP= AKBÓ

Hence.

akmas &KMB+ <KBC =KMBC

SKMP= ARMA

SKMAS KMBC

Proved

Join KC

· of ABCD

ABDKŁADEK FACBK ABDC = £ of #gram ABCD ADCK A ADAKT ARČK

of (DQKR + QCSK)

ACBKI A CSK + ASBK

of(cskQ + SBPK)

ABDK + 2(DQKR+QCSK)

<sko + SBPK) ABCD = 2-4BDK+DQKR+ 2 @CSK + SBP大

TE (DAKR + QC3k + SBPKYŁAPKR

BDK

APKR - QCSK

EXAMPLE 6 ABCD is a parallelogram

to BA cuts BC AC

AX Prove that ABCD is a Ugram

XYZ TA8

Given

To Proves -4AXY-ADYZ

Proof: Join Cz.

In Ai CX2,CKA;

They are on the same

my line pariet

As shown

point

on

Prove that

EXERCISE

diag, AC

ARD. =PQYB

P.Q. R. are points in the sides BC. CA, AB

Resp of a ABC such that POIFAG, ORII BC,

• ABP = 4 ACR

Prove that In A ABC

is the m

ABCD is a

LADC

ACB is a right angle,

-point of BC, prove that

AB = AD+3BD*

nadorlatival so that chec and each of 40" IF AP AQ

CB drawn parallel to CD

at P. Q resp., prove that

RA· AB = PA · AD

(生寫美旅一商王)勝幽之地密斯若國美

(海開堂會大在起日今展社申丙)

族

上課。多謝親友慰

·出院,休卽照常

「八八日康復

四一期出版

「今日世界」

影星舒耐熱,彩色版

10期出版,封面露

?册五角,販均 j印刷精美,冤安没 事盡,壯殘局研究等 精彩短小說,歷史故 院,卽造師住的房屋, 設計新的典克蘭博物 ,民航機中的巨無 倫台奇主張簡化其還會 市歌劇團的發展史,布 事件的政治作用,紐約 在努革命,巾醚共遶境 「營運南號,可蹤和現代

「平,北韓恐怖分子緊.

日內瓦協定與亚洲和 的原始藝術。自有

勝收的拍貼藝術,高山

曹之時,先後作腿術演講及示範 - 年,簡美國密西根州立大學藝術系之邀前在講學。六五,六六年,又應東

主君抵美五年,備受彼邦入氏歡迎,爲安揚中國文化藝術工作,買就甚多 宝用,並作弹棄表演,引起了數以萬計,美國,對中國盤均感興送。

·博陈言,及各大學不下二十餘究,誰會中之中國畫部門,該會帶進王君 巫展者,計有杏名福变盤踞,密西根大學與安阿爾市政府辦小型世界

•博物館安藏亦塔特產出,觀衆爾獎不已。至於美國各地,邀君學行 *九六三年二月,香港大會體會爲其主辦門神年畫長袍,哄動一時。英殿名 史收藏中國民俗藝術,米麵,石刻,碑帖腔:尤門神杜君木湖社名。 著作品趙倫、特在中國畫班,王君主持識投,王君除創作外,

【蘭市密州立大學,津區大學,與馬上被等。

拙,有鹊卷氣,知名海外內。會被選爲香港中國美術會副主席。一九六四 盛行。會風遊國內外名山大川,他研古畫而不滿於古,養功深,用雙斲 王商一小傳並受酸家王商一,廣東中山入。早崴,在上海爾: 一京院等地,從事文化藝術工作三十餘年,披髙山水

童軍總會主辦

有心民

述之活、 日八期五·在黄大仙肚區中心,及元朗難杀局中裏分別無晚會。L 第〔星期四)“在柴軍紀念營地及粉嶺大論中心分別熱火。四月 编程算诔部,舉行營火會,茶青海厝街共灣官立中學舉行游痍股食。四月一 *水喬柏雨中學分別舉行火食,四月二日(星期三)在九髅加士居道」 福利中心舉行耐火會。四月一日(星期二)·在米西環仁法院3靑山 | 大埔绿民府官馬後無主持新界地世年過之開幕禮,在荃灣城門道天主教 * 硖尾嶺南同學會小學,舉行業藝會,在大堆王朵枝中學舉行號类商;並由

領統會主辦之宵年遠,昨,箸之實際收益(1) 七時中,假荃夏城門道,大埔會在,合唱。(琳) 薏 嗚。此實該晚齊參加 荃灣特安,-

份使青年生共| NDS 欒像演觚流行音 活動最一連八置在仇體伊利沙伯洋館舉行一個同激晚會,招待二

·軍廟表黃柔道,第一見習銀袖旅大合唱,及蛋年杯籃球冠軍在 小學大合唱,威靈頓英文黃院童軍樂隊表演,成人教育好給表性,並由 之捽角幾集,荃灣官立中學舉蹈表演,惠州槃她舉行婚蹈表演,元朗官立 R THE SOUNDS 槃之養,裘錦秋英文書院之槃除穢奏, 青年 用,計有居喀兵團之樂像慣奏,男拔萃之西洋湖,與女中之聖彌表演, 演,第一見色版社峻穴,

·童用議席家體剪綵,並展出各活的鍋片及資料。同在后 ; 整序:三月册(星期一)下午六時,在九張覺士道議會聽眾,畢行開

·七千餘名新界悲九之青年男女。骸嫩大之青年週活動,由純青年訓練部 一個晚上。分別推出一項艱大的活動計劃,命名爲青年遇,預算招待爲歡約

·二特瓿心童軍耪戴於復活節假明(三月卅一日至四月五日),汪 去招待各地健之骨年。六六之

招待港九新界七千青年

靑年週共六天

天主教福利中心舉行一

街道昨軍射

·歌唔緻呼网烕土風舞等:區之三百五十多位年 而在營內會中有短劃表一式中,并與來自地 舉之意軍饵補惓助進行。官業主賽場森德在 持,并由多位有經驗,縱藝晚會。大埔理民府

醉在脂韻藝火中。會後,晚會在儀式後,節目 逐開!竽海足蹈,隔一年青年在社會上之查任 一富,面有··居喀兵

年答火大會,招待當地,就現行之青年團,昨 的三百五十多位古年势女一晚七時,在大埔 轉。由青华爾抽徐嘉黃主任、枝力图雀常,舉辦國畫名家

年

臺:近衆多

山水大

梁伯康復

適,本月廿一日

勞神過度,感覺不

*生悉心檢查檢? 院作期 。經

無表演 THE SOU 並無不良狀況,於