·ABCD is a parallelogram and HK 13 &
日七十月二年午丙夏廢
WAH KIU YAT PO
華僑敎育
Example 4
九六六年度
英中会考
一九六大手 試題預習專欄
數學科
(十)
歐陽錢文·
JA
MATHEMATICS (10)
LESSON ES
IMPORTANSY THEOREME;
↳ slid - poist Cheaem:
TRIANGLES I. Ma-P. & INTERCAPT.
THEORENA.
SIMPLE RELATIONS Between The
TRIGONOMETRICAL RATIOS.
The st. line joining the mid points of two sides of a a" is'
car parallel to the third side, and cbs equal to # of the third sid. 2, Intercept Theasm.
The st line drawn through the mid-point of one sida fia a
bisects the Mus` side. pakallat vo another wick
Or, we may generalize the intercept therem as follows:
If three or more parallel st. lines make equal intercepts on a given transversal, they mate equal intercepts on any
ather transversa!
AB
SOME FOUNDAMENTH RELATIONS BEINCE TRIG-RAIRE :
↳ Reciprocal relations:
ast&-sh
:
CASA
@
tan 8= casa
Sing
当
Sin*) + Cor*p =1,
Dec2 &=1 + Lañð,
cse2d = 1+ codeD.
st. line outside the ligram AP. BE CK DS
are the perpendiculars from A, B, C, D 4, HK Plove that AI + CR = BQ+DS.
Proof: Let AC cuts BD at 0; dres
ONE HK
Sing APRC is a trap, with APBOvť -
·0 if va mid-pt, of a £ 4 ́'onlarUER (
• one is the median of Trap, APRC H. V. ONL.CAPFCR)
P
And in drap D5Q6, ON =Ź. (DS + BA)
Ź (AP+CR) = + ( D $ +BQ)
2. AP +CR = DS + BQ
1
Example 5 Express all the trigononchical ratios, of ot in terms of size)
Saluto:
Sina + Costdat
Tam of ====
Sec of=
Csc of
Cof of **
Casa = fir $4
Sin of
Sin o
Remarks All the trig - ratios can be expressed in terms of any other
Example ! If P. & R. S are the mid-points of the sides of any quadrikaters! ABCD. Phose that PARA A
Proof:
a
parallelogram,
Join AC :
**$*
* PRIRE
Mid-pt. Herr«.
RS AC
Similarly, in SABC,
PO LLAC
Consquently,
A
R$ £ PQ (# LÁČ)
.. PQRS is a #gram
>
two sides
C
Example 2. Prove that, the st. line joining the mid-points of
the diagonals of a trapezium is paralled to the bus and equals half the difference of the two bases, Given: Trap ABCD with AD/BC & AD<B¢
AE FEC BFFFD
To Pove: (1) EE // AD / BC-
Q) EF = (BC - AD)
Proof Ket & be the mid-point of AB.
A
Join GE. GF (we will plone that 6, F. E are collinsar.)`
In ABC.KG, & are mid-yt. of na, ae map, (streng
Example: C. Given
Solution
Cas A find esent and cot A
Cos'A
Sin At
¡Note It is always possible to describe a it when two sides are given.
In the above example, we can constract Hayet
St. ACHS, AB=13, from Pythagoras, BCE,
CACA = AE =1, cotta
Example I
Prove the following identiti
Proof 'casicot *ex - 1
= cactoc ( cactus ==)
z csorbe
(1ɔ (tan # + fac 0
(+)
@
-
日八月三年六六九一曆公年五十五國民集中
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四第消尙
息有
張刊敎
在育
pt of AB, BD
mis-point
i Since
That
AD/BOX
-
EXF
are collinear S WHY 133
BADA
=Ź(BC-ADI
Note: (1) You should complete the proof by adding file Cake ADDRE
then you will find, the 2nd pt to prowing.
You may prom
this problem in more one 'nmy.
BC at H. R. Join AF and produce it to must BC.
Then AFP *H CB (ASA), bance E. Parė., mid-7), of silke M DANCI then EF8&c and
Excele 3, ABEF, ACEH are squares outside SABC. AM
is a median "and" produced to meet Fit at &'. Prove that an MA & PH (2) AMAL PH
Proof Produce AM to P
5.1. MP= AM
Then ABPC is a Ngram
Cding, biset task offer).
<ABP +<BAcma di Cintike B#3
But «FAHTRA 21, 2
Andr
ABP¿FAH
AB- AF
NO PO
· tau &- catß.
tana- catß tom scot a (b) (sina cup + cosa amp)+(cosa cup - sill
sxbople 8. Prove Hut (as ta
Leyote
cots
tawa - estß
Land Lot p
£. H.S = [(sine coups) + 2 sind coss-cara sing + (cosa sip)*]+ [(COLD COB) =
a coigter sind sist
+
x and B are two angles and quite independent of each other
Hure &
Is (as hxp
-
(d) Exp
10) Exp.
But
BY = AN
猜
preted,
(Br30 FAH,SAS,
AP FH coni, side: 57 zaj
AM
AP = FH
f and BAF - Hay
* The M. F. &
(hi Ep = (x+ 38 mm
Ratto in speed = Train for a 45:36
Rate the bank pay!
-
25.38. I SATO
The • man receives
liabilities
The rate the man pays
As asset
-ANS
His liabilitike
5. The ratio shi FL.: half-crum = 1:2:
6 *
10%* - 21X − £0 ={5x+2)(2x-6)
ANS, CURE sach
*. If 52+2 is a common facta, then tool−†)*+ 10 (−−) +8=0 | | 22- (ao (+£)*+40 (+5)+p=* £==
r 2x-5-
· · * x + y + z = + + + + 2
z(x+y)+xy (1-z) => (Z + xy) (1-2) = o
14
But x+y=1-2
then either 2=xy ov z = !
If z=-xy then (x+y+ z =)x+y −xy = }
Hena (x-1) (y = D) = o
8, The ratio in
B receives
www
capitats
= "
=f3500:£5 17:0
1 of the remainder of profit = £450,
惜
To at the remainder - £450x
7.=6317
A's salary_m_t:do - £317 == £372.
Exercise SB
1. In fight is the mid-point: af BC; PAQ
{
a st line such that AP= AQ. AP = $(ASTACI
Prove that
prove
that DMAC.
I, IN DABC, LC = 2LB, ADLBC
If M is the mid-point of BL.
3. The diagonals AC, BD of the square ABCD intersect. at k, the bisector of LBAG cuti Bh_at_ K and catar "BC at Y Plove that. CY=2kx.
2 in 1 - 3 cons
4. Given & Fun A = find cos A and CAC A 5, If sec a = find the value of a sind 4 con 6. Plore the following identities.
by cotro sees -
=
C, sinoc tant + core cota + 2 ind cord = ton of 4 Co
d, sin a carts
FH
Page 15Page 16
七十月二年午丙胶厦
WAH KIU YAT PO