1968-04-09 — Page 14

貫二弟張四第

日二十月三年申戊麼優

WAH KIU YAT PO #

B

墓

期墨

曾九月四年八六九一曆公年七十五國民中

168 英文中學會考試題預習專欄

{*** APBQ is

a rectangé:

PQ = AB

數學科

(廿四)

歐陽鋊女

MATHEMATICS (24) LESSON 24 CIRCLES

Example à l'on chords of a cucle) *. Y are the mid-points of the Chords AB, co of a circle centr

0; XN, YM are the perpendicular from X, Y to CD. AB resp If XM

cuts YM at P

prove that op

and xy bisect each other.

PRODE: Join Ox, oy

A

M

x is the mid-point of the chart AB

OX LAB

Since YML AB (given)

PROOF: Since PE, PQ am tangents from P to DA

AP biscots «EPQ Similarly! Ba bisects « PQH

« EPQ = « PQ H

· p = q 2. AP # BQ

[R14.25" GHIEF

att, cs, p=8.

Similarly," AQ, BP are the bisectors of act 43 901, QPF

Similost;

By def.

OX #YM both of In one perp. to CD, OX #PY. anot DY //*!

OY //XN

Qar

APH BQ

*. AQ H ́BP

and APBQ is

AQ #BP #gram.

**A*

48#5%$#$%$# $ # $#%$%$#$#%$%#$%$#$%$##%#$#$%^»$#$%$#%$#$%3 #3%»S

¦ 345%$#543 #ESÉS # BAÆS #53#563 #373 #$%3 #8763

僑

が完

おか

| #$%$#$%$#$%# PELS +5%S#$%$%$#$#%#%#&%$#&%$#&%$#3%83%5/3&#53 +

*#$%$#5%$#$%$ »»5%% #$%$#$%3 #5%$#SFS #3%$#$%$#$%$#$%$#$%$#$%3 #

1968

類習壽酒

歷史科

(廿四)

·嘗一民,

中國歷史科預習問題答案

第七章

第四範

宋朝

北宋的衰焦及未變的南波

ding, of a rect

(十六)選擇題:

1. B

1. C

(十七)填-

題:

1.(A)漭長二年(公元二七)

(B)**

1838 (5)

2.(A)脔天津

(B)

3. (A) Ta

(B)恭

(十八)試分忻北京己上因。

46 BQ on the bisectors of two adj. <s en st. bae GH.

• ARB = ÷ of a des = {a APBD is a rectangle

AB== PQ

PYOX is therefore a ligram Op and XY bisect each othe

Example 2 (Angle Properties of Circle... The bisectors of the angles "ABC, ACB of a ABC intersect at I und cut AC, AB of Y. 2 resp, the circle,

812, CIY meet again of X Prove that «YXZ +<B!C = &Hus FRODE JOM EX, with the notwbians

8,

as shown in hig

in some segment

C,

<7* * - b,

C. CA

b. mb2 (gimn>

<Y*2=EXY + 2732

mt, by

From 18C < BIC + by

Example 3 (Concycké poink )

ABCD is a #gram

O is a pit inside ABC D *.* • AÐB⭑¿COD = 2 É A Prove that.

40BC = <PDC PRODE: Drive APHIDO, BRICO

Since the three sules'of oHBP Qre parnitet do the three sites, of a cao

But

La ABF in ADEO

AB- CR

ARP = &DÇO AS4

C

This live may not Zpass the Zithe ́pt. of inter, of

(< Sum of 4)

Exampk_1: (Alternate Segment:

ABCD is a minor arc of a circh such that AB - BC, AB' and DC meet, when prodead, at P, and DB is produces to meet the tangent AT at T Prove that TP - TA

PROOF

: AB - BC

given és stand on egoat shouts are

AT is a t

1 J. + ABC

▸

A

答:北方四大湖開鎖,初期尚可見作易,但遠至瀾,名號,卒於清康二年微飲二帝慧金|

人巖了你,共联战一百六十八年(九六○二七)。其亡之因,有下列各赠: 1. 外惠想 入球统一之後,國土爲絲燈確的。蔬宋初未能收碟石敬请手 ,形撕防上一大漏洞,泺寫務所乘。此時遼人未服,西邊文考西買 之發物,使中有前後之要,後來源锥气卡金聯兵所亡,但隨之無人很大,代途而成東 北之鬣。卒於北宋亡在金人

2.

大宋之對外政策,只知始展以誤觸,如蜜洲之役,宋兵造凫御金,軍 高清者,本可擊敗兵,奈眞宗思憶,與遼和,而有澶淵之盟訂立。後來西夏及 加入之入要,又未能考於處理,竟約金人降將螢殻,金人乃額此人便,匯成「靖康之

]

#

言时只因 一由於冗兵,冗吏的消解,及對外用兵的支錡, 因而發展•彼况衛書

密支付、夏、魚的錦輝:國際在此躍大的損下,財政駕得不困。

——自于安霾雾相之後,墾中大夫,需了彼此對政局的成見,形成新瀾 黨發、新黨以王安石、承惇、曹布、蔡京無食,而畫業以司馬光、密竊、蘇軾等篇 質。初尙骂了彼此之政見,及安石崩相後,而辦於意氣之爭,一黨得勢,便排斥贏贏

*蕉四十餘年而不息,本與朝政混亂,演耙张。

4.

Lt. sigment

ime segment

sides

Exemple (contact" of "circles)

AB = Þ.

AC = 1

*BAC * fo*"and big 15 the min-pt. of BC. Circits art drawn with 98′′ and AC as diameters? Provm that two circle. can be drawn with O as centa to touch each of these! (Cirottsjand find their cadir, in terms of and ta

(+)***ASONZES) •

答:一、南宇之進寺——他人撰微鉄二一法,乃次螢邦甚爲楚帝,欲以漢制漢,但為臣所贏

· KAREEN • VENDTABAKER=ARX · ****** (EM) › PERSPE ,下開南宋個安之局,此時北方則需金人所撰

二、宋金和刷一遍翩初立之時,任用李箱等相,主要「能守認後能,能觸而後 點

★一時頗想作爲, 奈爾後來高字僧黃潛等、汪伯彥寧願和之擬, 謝烏金人所乘:乃大 無進攻,識金陵,高宗乃奔臨安。此時華韓世忠及其夫人樂紅王於黃天萬大敗金兵,使 魚A因此不敢前含工。不久会人又對象關機,主理河南、陝西之地,以作未会之耀 頓 ,但行有邊整理北方*同時及故意放案牌證,於是備和之又起。其後金人主驗量| *用心發政,讓和中止,乃於紹興小年(公元一叫〇年)再大軍南侯,此時江南軍路 新起來盟瓣義勇軍抗藏,尤以飛之軍勇,於係城(湖北襄關北)大破传兵,並興勝

「趙繫,高達朱仙缱(河南開封西南)正欲攢日渡河,官搗黃龍府,以深二聚、無奈釃和叉 会社

三蔵。 十二金牌谷岡京・後露以「英樹」母飛死帯 「村風羊集和——岳飛死钱,和送成,乃於解關十一年雙方訂約,以大散關為未飲 我長,辛場網臣於念,並常就其金,鍛、蔣合一五萼落聞。未只得回韋太后及微宗之

依據兩宋之形數,求無可觀之將,而建打此品等條約,置乃無療內心隱私。君使南 人心載大受挫折。

中國歷史科預習問題

"Join Of's and produce it to cut DE¦aƒ ̈H K

T OF

When OF AC, & mid↳ point theorem)

£#*EA = ±AB

*OH = ABMACIJE

Similady, OMR OF IFM

AB -- DOC

AQS= – DOC +

Com as

- 408 - ara

Jan Op

O AN | Concyali OC 4 PA OF # BC

Const.

provat

✔

prove

0

Example 1 at chads è Ares)

à ABC inscribes in a circle as shown, D is the roid · pl of minor are BC. O is

A

paint on DA »t. DO=DC Prove that, O is the in DEMİR, of 4 ABC.

PRODE· Join Oc

Minor arc BD =

SUR O

prc.

DC

Given

In the same circle, equal ares subtena igual as las õuj

2,

*

vie DA is the biscotai of BAC

It is required to prove that OC is the bisecto of «¿.

In aon't

\* = 0, + C,

HONGKO

PROOE Let E. F be the mid-pt. of "AB"

"AC mǝp

basra isa o

< DO=DCj

BCA ---GEON, Conclude that

Im ADCO Back

OC is the bisector of From-ai and abo

Example $• ABC is

O is the in-cente of 2ABC

an equilateral a

inscribed in a cireto". P is any poit ca minov are BC, Prové

May

PA = PB+ PC;

PRASE: Prostace BP do 1

suot that

SACPD

PD=PC PC=PD

Const

« CPD = « BAC =60°

chet, a of Cyake junts

2. APCD is equilateral

Consider — APC, BCP -

AC m BC

<ACP = « SCD ( = 60*+ = }

AACPE ABCD

NOTE

wher of thв and FM=FA = £ 40

OM CAB + AC) -.............

From cas and «by, we have « OH = om

with gente o radius on-hik Can Heaw one such circle

msider now,

ON = RF - NA

Hence, with cente

ON and of

radus ok #tip-js

we can draw another rege circle,

As shown in Figure (to the left)). CAMABC is said to be

circumscribed about 00

The Quad

LMNP is said_to_be]

inscribed in

<C> ☺ o is

☺o.

said to be

Circumscribed about LMNP

• ABC

DV

inscribed in

Hues & Aves. To Ex

Then

a*« <p ̃ng+y* = p* * 2p′′g" + gu

[

11, Methet A: Ket amp' + qˆ. b=2pg.

Whea

= +(AB)(AC)

ABC "is tot n

[Merhad B By the area pormula. ftf«a}{3=b}{5 «c)

where

sides of equi a POD

1A/

[Ri Jon AD

AP = &D

swa Bb = B P + PD = BP + PC.

AP = GP+ PC

NOTK w may take a point E on` AP such that_PE=PB

Hence

prove

that aBPC мA ÁBÉ (4&PE

aquilateral), then PC = AE

Example &'( tangents)

As shown in Fig 9,8 ast

is

3,

As parastal lines Gün and EPF, the common tangent

rà fouchés Euch circle

centis of two ciretas touching

as shown. Prove, that

8D = ÷&C

GABA -Źof a #BC × 8 The dist of D fram AB =

Similarly BACD =††

- Ca

the dust. of D from AC = 165km 32 cm

(0) → ABC= APQR = {÷APC + 48¢#

+ LARR) \= 470 - H4x5+20/+3an)

* 17 1⁄2 Cap J 'ham # >8?c. by Pythagoras, A==√P+3* *5 (5.#, i... ant

Roth, 1905-)

第七章

宋朝

第四齡-

第五節

北宋的衰亡及宋室南渡 宋代的學術

thin

Wy:

***AE÷FALA)RAST; (B) REC

C)蒙古嶼】。

1

魚宋帝鷗踏海殉國的忠臣(A)天雅(日)年(

C)秀夫]。

2.

FITURE:

成吉思汗部位以後,即大帶征伐,先發歐三次征,再滿

(A)及(B),然後入主中國。

1. (A)

(B)

宋自(A)關鯛,全都(B),迄微錄二步被據,共有一 百六十八年下,力(C)

2. (A)

(B)

(c)

3.

自高宗迎立於南京起,迄(A)蹈,共 有一百五十二年天下,史(C)

3. (A)

(B)

C)

(廿二)何謂帶廣之難?

(廿三)常宋如何理亡?誠路述之。

(四)宋代理學興盛之因素為何?試說明其效。

世界歷史科預蠲題答案(廿四) ·林希靈 · 芮組 歐洲 (二)經濟及社會的轉變(主要在英國)

(-) RE:

1.

A

(二)檢堑:

1. (A)

2. B

(CROP GATATION) (B)BERR (未完轉入第四張第三貫)

By mid-pl. theorem. "The line joining the two mid-pt. is 14 PRODE

ACD ACB

Vnd 24.03 = 324 324 Nytam XQRD.

•A¢â¤ ̧M + 01+ figrum QPBY AISA DE and AÏ-* AN

#gram XQRD = Agrani QPAY

Exercise 24

SAC - 31 ↑

R

C

1. A triangle ABC is inscribed in a circle, and” the

bisectors of the as meet the 04 « X, Y, Z

{.

12

+++

show

that the as of axY* are resp. 90°-4. 90°- §. 90′′- $

Two circles meet at A and B. AC, AD are diameters, of each circle, Prove that C, B.D are collintar.

3, Two circles meet at A, 6. CD is a commendwegerer"}

to two ciroles. Prove that «CAD+« CBD = 180"

A ABC is inscribed in a circle O is the ortho - centre of a ABC .` Tæp three altitudes AD. BE CF are produced to meet the circle again af G.H. K resp. Prove that O is also the incente of a 6Hk. 5. Two equal chords AB, CD meet, when produciet, at)

G. Prove that BG = DG.

6, ABCD is a cyclic quad. If M, N, P, Q are the

centres of four circles inscribed in 4. ABC, ABD, "ACH, BED. Prove, that MNPQ is a rectangle. 7, The diagonals of cyclic quad ABCD cut each other at st. « at Po Prove that the L from P to BC bisects, when produced, AD. (Brachmæg−)

upta" thewem)

8, AOB, COD a twos 1

Two chard's CP CQ

diameters of a circle cut AB at H, K. Prove that¦

\H, K. Q. P. are concyclic