百二篥張六第
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170英文中學會考試題預習專欄
堅道英文書院主編
數學科
(十六)
MATHEMATICS (26)
Example 1: (on chords of a circle).
X, I are the mid-points of the
chords AB, CD of a cirole, centre
O; XM, YM are the perpendiculars from I, to CD, AB resp. If XN cuts YM at F, prove that OP and XI bisect each other.
Proof John OX, OI
X is the mid-point of the chord AB OX LAB
Since "YM L AB (given)
Similarly, both OY, XN are peri
*OT // XN
OX//FT, and OI/XP
By definition, PYOX is therefore
OP and XY bisect each other
Example 2 (Angle Properties of
The. bisectors of the angles. ARM
of AABC intersect at I and cut AC, AB at I, Z resp., the circles
"GIY meet again at X
BIZ
Prove that
/ YXZ+ / BIC - 2 stil Proofs Join IX, with the notations as shown in fig. Ds. in) Bame segment
IXY
By
circle):
АСВ
-(gaven)
gram
M
IXY + / IX2
From AIBC, BIC +
/ BIC + / YXZ - 2x
Example 31 (Concyclic points)
ABCD 18 a //gram
Dia a pt. inside ABCD
COD - 21.
B.t. LAOB + OBC - Lone
Prove thati
Proofs Draw AP//DO, BP // CO
Since the three sides of ABP
are parallel to the three wides
of ACDO.
ABPDCO
nut ABCD opp. sides.
AABF ADCO ASA
APB DOC
/_APB + __A0B - DOC + ₤403
A, P, B, Oare concyclic Join OP, OC: P3
-A
nstr, å proved] (PPUVE)
(a in mane segment)
Where a
報日橋
AO BO
ACP
-
CF - CD (sides of equal PCD)
CAP ABDO
_BCD (~ 60° + α)
AP BD
Sinca BD
BF
PDBF PO
二期星
AF BF + PC
NOTE We may take a point & on ar suon that PE PB,
Hence, prove that ABPCBE (ABPE 18
quilateral), then PC. AB.
Example 6: (tangents)
As shown in fig., A, B are centres or two circle touching the parallel lines GQH and EFF; the common tangent PQ touchen each cirole as shown. Prove that (a)APBQ is a rectangles (b) PQ = AB
Q.
日八廿月四年〇七九一公年九十五國民窖中窅教僑華
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10英文中學會考試題預習專爛
堅道英文書院主編
英文科
(十六)
ENGLISH LANGUAGE (26)
Answers to Exercise 25
1. (D) 2, 3.KE)
13.
25.
14. (D
26.
15. (0)
27. (E)
4. (4).
16. (0)
28. (D)
5. (DE
17. (B)
29.
6. (D)
∙18. (A)
30%
7.
A
19. (A)
31.
18. (A)
20. (E
32.
9. (E)
21.
33
10:
C
22.
34.
11. (0)
23
35.
12. (8)
24.
Proof: Singa PE, FQ are tangents from P to
AP bisects / EPQ..
Similarly, BQ bisecta / FQH
LEPQ
=
/PQH alt. /_1, GB//EF
AP//BQ alt. Los pra
Similarly, AQ, BP are the bisectors of alt. GOP. QPF.
AQ // BP
#F//BQ and AQ//BP
APBQ 18 a //gran
AQ, 30 are the bisectors of two adj.
line GH.
LAQB – or ? rt. 8 a 1 rt./
And.
IPBQ is a rectangle
diag of a reces,
Frampls: 17: (Alternate Segment) ABQD is a minor are of a circle. such that 4B BC. AB and DO mest when produced, at P, and DB ia produced to meet the tangent AT at T. Prove that
TPTA
Proofs
AB BC given
/ § stand on equal chords are equal
AT is a tangent at
L
ConcyOL10
LTPA - 2 L in
TPA -
@egment
Begment
TA TP sides opp. equal
Example Ŏs ("Contact of circles);
In
ABC, A = P, AC = 4, / BAC 90 and p q
in the mid-pt, of BC Circles are drawn with AB an AC as diameters. Prove that two circles can be draw with 0 as centre to touch each of these circles. -an find their radii in terms of p and
Exeroics':26
Choose the best answer to complete each of the following sentences 1
The white dog is as clever as, £7. not oläver thas, the black one,
(A) the white dog 18 cleverer than the black dog (B) the white dog is not cleverer than the black
dog..
C) the white dog may be bleverer than the black
dog. Ben
the black dog is without doubs NOT: RO CLEVET as the white dog
5) the black dog is
the black dog is cleverer than the white dog
2. Nearly all his friends were convinced that he was
a clever man. The manager, however, did not have full confidence in him, This means that........... (A) all his friends thought that he was clever
olaver
(C). the manager could not rina enough evidence
in support of his capability
(D) everyone knew that he was clever
(F) one or some of his frienda were not sure, ha
was claver or not
But for his generosity, we would have been puti into prison, This means that.....
(A) we were out into prison for he did not help
(B) he helped us but still we were put anto Ni prison
(C) he did not give us enough nelp so we were
put into prison
(D) he gave us enough help to save us from being
put into prison
(E) he gave us much money for he was generous
he gave u
He needn't have been so anxious about his diater
who, in my opinion, le cautions enough.
(A) The carelessness of his sister makes nam
worry much. Y
(B) Ha does not feel anxious about his sister for
he knows that she is cautious
(He feels anxious about his sister for ne
tahe is cautious
doesn't think that.
(D) It is not necessary for him to be anxious
about his sister
(5) It is necessary that he feels anxious about
his sister :
1 was to have paid a visit to my aunt in. That case, she wouldn't have been so worried. This... means that
(A) I visited my aunt'
(B) I visited my aunt and this made her worried (C) Ky aunt was worrying whether I was going to
visit her
(D) I did not visit my aunt for 1 alan ́t want te
worry her
(C) I did not visit my aunt and this worried haz
Example 41 (chords & Arom}
ABC insoribed in a circle as shown. D is the mid-pt. of minor aro BC.
point on DA 8., DO DC
Prove that 0 is the incentre
of ABC
-Proof; Join. 00
minor aro BD - minor are DO (Given }
In the same circle, equal arcs subtend equal /a
, 01 in the blsector of A BAC
(at Sca)
(*)
It is required to prove that 00 is the bisser of
LC.
In A0101:
IN AUGUS
EXT. L OF L
Dasa La 1808, A (10«DC)
in same s02,
OC la tha pissotor of / BCA
From (a) and (b), we conclude that
0 is to in-centre of AABC
Example 5: ADU 18 an equilateral" inscribed in a circle. P is any point on minor aro BOL
Prove that PA - FB + PC
Proofs Produce BP to D
such that PD
PC - PD constr.
In ACPD
|_ CPD - /_ BAC 60°
(Ext. of oy olio quad.) APCD is equilateral Consider As APC, BCD1
PC
Prods: Let E, F be the mid-pt. of AB, AC resp.
NOTE:
Join OE and produce it to cut OE at H, K Join OF and produce it to cute Fat M, N
OH = 08 + EH
Where O5 - AC (mid-point theorem)
EHEA
ABA
OH = ♣ (AB +AC ).
Similarly, OK - OF + FM
Where OF ZAB and FM - FA - ZAC
OM − 7 (AB + AC)
From (a) and (b), we have OH - OM Hence, with centre 0, radius OM D we can draw one such circle. Secondly consider now) ON and OK OK = EK - EO • †(AB ~ AC) - 4(P ON - OF — NE - KAB - AC) # *(p= Hence, with centre 0, Tadius OK -
we can draw another req'd circle.
As shown in figure (to the.
(a) MBC is said to be
circunscribed
0.0
(b) Quad LMNP is said to be
inscribed in 00.
(c) 0 0 is said to be
circumscribed about LNNF or inscribed in /ABC,
The brave prince would nave detested the giant but that he was sick at the time.⠀
(A) Though the prince was sick he defeated the
giant
(B) The prince was sick and ne did not defeat the
gianto
(C) Though the prince wes not sick he could not
defeat the giant
(D) The prince felt sick for being unable to
defeat the giant.
(E) Though the giant was sick at the time the pranes.
did not defeat him
7. It is doubtful whether there is one who will not
be greatly surprised to find him unable to pass the test..
(A) There is no doubt that ne will not rail Th
the test
(B) Everyone believes that he'll fall in the test (C) No one expects that he will succeed
There is only one person who feels surorised to see him rail
(B) No one expects Dist he will not succeed.
o one
During the foggy night, many a prisoner escaped from the island without the notice of the sentries (A) Only a prisoner had escaped from the island: (B) The sentries noticed that a prisoner had
escaped from the island in the foggy night. (C) Some prisoners tried to escaped from the
island in the foggy night
(D) The sentries could not find the escaping.
prisoners in the foggy night.
(E) Many prisoners escaped quietly from the island,
in the foggy night
I could hardly finish my work on time though I had tried very hard.
(A) I did not finish my work on time for I did
not try hard
(B) I found it impossible to finish my work
though I tried hard
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