買二弟張六第
日五十月三年午丙春夏
WAH KIU YAT PO
華僑教育
***A
試題預習專欄
數學科
( 十四 )
歐陽鋊文
MATHEMAILSSS(4)
LESSON YA CIRCLES
Example 1 Con chords of a circle)
X. Y are the mid-pt of the chords AB CP of a circle, centre 0; XN. YM are the & from X. Y to CD. AB resp. If xx cuts
YM at P, prove that of
XY bisect each other
Proof: Join ox, OY
*.*
M
A
x is the mid-point of the chord AB
. OX LAB
But, given YM LAB
Similarly, both of XN are
2. OX HYM
CD
OY HIN
'. OX HPY * OY #IP
PYOX is therefore a igram OP XY bisect each other.
By definition,
Example 3 ( Angle Properties of a circle ›
The bisectas of the angles ABC, ACB of a ABC intersect
at I and Cut AC, Að
at Y. Z resp. the
circles 812, CIY
期星
日五月四年六六九一层公年五十五國民中
育教童
Example 3:
ABC 13
an
equilateral s
inscribed in a circle
P is any point
arc BC
Prove that
PA = PB + PC
Loof Produce BP to D such that PD=PC
In ACPD
PC-PD
• CPD = BAC = 60*
Conso
K
Ext. of cy clie quad
Consider < APC. ACE
AC - BC <ACP = B C D (~60 + x)
СР- Со
SAP ESC D
... Apa BD
Hence, with–cente 0. Aadics Ok=Ź(P-8); we
can draw another circle to touch each of the
given circles.
HINTS & AUS. TO EXTA
and
The tima. laken
Oct. 15-y
! Since the cost of bill is equivalent to the Princ, in S. 5.
the banker's discount is equi. to the Interest
Sept. cao (of the same vi.) * grace's days. - 28 days 3 yr.
2
23 The bankers, dù count. =6% d£ £389) × 365
Ok...since I=PRT =
x% xx
Sides of a PCD.
5.AS
NL GAYS T = 3656ays
3£{px2n1] x[ 14 £*£% + 1)÷ 10* which is what we called the "THIRD - TENTH -- TANTH - ROLE" Discount <= £ (3 8 5 × 2 × < 774)× (1+ 4+*+20=105
Since
BDBP + PD. – BP + FC
PA = PB + PL
Note: you may take a point E on
AP 5.1 PE- PB: Then, prove Mat A8PC = ABE (mote · A BPE À equi), then
PC = AE..
~LLIS
; by practice mrkod * 3 s id.
217
Example 6 - Tangents)
The truckTM present worth The truce discount
The time taken
As shown in fig. A,E
are the centres of two circles touching the parallel lines GOH.
I
The rate = PT
3,
EPF; P Q touches
each circle Prove
that (*) APBQ is a recr.
b) PQ = AB.
freef
Y
AP bisects
Since PE, PQ are tangents from P to 0.A
Similarly
Bit
e a black LEPA = PQ H
EPQ
Pan
GH JEF.
zee... P
=
p = £786 -£30 = 750.
=I= 30.
- T = &mth, m gr.
67%. p.a.
ket to be the principal then tax bɩ the amanRT as At Simple interest i Interest = 221-2x = £*X 57XT
.. 7 = 20
b. At Compound interest: Amount - £ax = £x ( 1 + 5%)*
.4, (a)
Subst. 2=
= {(ag-1) inh_*** 517 +35*+340. then домогу навалунах (урт) =,
ANS. (-3,-4) 8+ (-5-7) (b) From (2+ y) −x*+ y * ~ 6 (X-3)*=0
we have [(x+9)}+=(x-9)][(x+3)=3(x-5)] = Hence either 31-4=0 o zy-x=0,
Ans. (-1£ ~*) or (2, 1)
meet again at x B Prove that
zYXZ+ 481£= aƒ«.
Proof Join IX. with the notations as shown in fig
<IXY = C,
Gar
some segner
112 - 6,
Since
<1 = C
and by - B2
given
120
In
A IBC
ZX Y + - IA & =\Cy + bz
< BJC + but
Similarly. AQ, BP are the bisectos et
the alt. <: GQP,QPF
...AQ #BP
40.1 BP
AP Ba is a Hgram
AQ. BQ are bisectons of two adj. as
Example 7 (Alr. segment,
(cirere
and be when
ABCD is a minor are of
such that AB= BC · AS
produced meet of P. and DA produced `to meet the tangent B1 at T. Prove that TP=TA Prack:
* AB - BC
give m
stand on equal chods are equat.
Example 2
( Concyclic Points >
ABCD is a ligrans
o à a point in side
ABCD such that
-A0B +« COD=Ara
Prove that
4 060 - <ODC.
Proof: Draw APADO,
ONG KO
sides of amp are parallel to
Since the D
the three
"But
Sides
of a coo
AB = CD
AMPE ADCO
Dep siles, ligram.
AS.A
APB Doc
are Concyclic
Join OP.
OC LPB
DP # BC
Constr. Place of
But
Example + ( Chords & Arc
DABC inscribed in a
circle as shown.
mid - point of mina
are BC. D is a point on DA
DO=DC. Prove that ou
incentre of ABC
Join OC
AT
on a st. line.
shay of rest.
Let
sii all.
+
• TPA = 22-
same
segm
-TPA = x (« da)
TATP
sides øp, equal –
in same segn
Example & ( Conface of circles)
InaĦBC, AB=Þ, AC=8
<BAC -90. and p >8.
O is the mid-point of BC Circles are drawn with 45 and AC as diameters Prove that two circles can be drawn with o
the same circle, igual ater subtend epi
sector of 2 BAC
DAC:
from LA
conclude that
ISOKA
the dist. P Q = 1 mi.
spead = 2 mph & Yé speed they meet in
mph == mph
the after X starts. That is.
[the after & Hence,
I¤ ̧ 21(47 −3)
20
ANS, A+ Neo
Xf distance + Ys def. == which is indep. with
Y yo a the inner radius of the path.
I yd. the width of the path.
area of path = m[I+x)" - my*«@#!x+7x* cost for graveling the path = &(2nrx+1X3] 8h.
= 6
12.9.
Length of has edges =27 [k+7+x]=ancer+x] Cost for edging by stone = 21 (artx)· 3 sh=219_165.
Exercise 7 A
ANS. I loyd X= { »
A triangle ABC IS
sectors of the Is Stibed in a circle, and the bi-
the Off at X, Y, Z... show that the as of oxyz are resp. 90-£, Ja-m
Two Circles meer at A and B as shown AC, AD are the diameters of each
Plove that
are collinear.
"wo circles meet at 4,8
Prove that
CD is a common fangent
AABC IS Mscribed in
as centre to touch each of 4 their radiizin terms of t
circles - and find
of a ABC. The three
cides:
PROOF
Let E, F be the mid-pl. of PB, AC resp.
Join of and pro
JOIN OF
where OE-
••-OH =±(ABTAC)
Similarty 10M = 0F +FM:
Where OF = $482&»FM=FA®£AC,
OM START AC)
From (a) & (b), we have
DH = OM
Hence, with cemte 0, racine, OH= £(3+8), AL
CONSIDER WIE
OK FEK - FO
==(AB-AC)
CAB-AC)
the
produced to meet the circle again Kresp. Prove that: O is also the
Two equal chords HB, CD meet, when
Prove that BS = D6.
at 6
ABCD is a cyclic quad,
of four circles ins
Prove that MNPQ IS
Centres
ABD. ALD. BED.
The diagonals of cyclic quas. ABCD cut each other
af yt = at P. Puve that the £ from P to BC
when
pindisced AD C Brahmegupta Theorems
AOB COD AIR
I diameters of
Two chords CA. ca cut AB at that H K. a. P are concy clic