Maths

TRIANGLES 143
Example 9 : In Fig. 6.43, the line segment
XY is parallel to side AC of
ABC and it
divides the triangle into two parts of equal
areas. Find the ratio AX
AB ✁
Solution : We have XY || AC (Given)
So, ✂ BXY = ✂ A and ✂ BYX = ✂ C (Corresponding angles)
Therefore, ABC ~ XBY (AA similarity criterion)
So,
2
ar (ABC)
ar (XBY) = ✄ ☎ AB
✝ ✆
✟ ✞ XB
(Theorem 6.6) (1)
Also, ar (ABC) = 2 ar (XBY) (Given)
So,
Therefore, from (1) and (2),
ar (ABC)
ar (XBY) = 2 1
Fig. 6.43
(2)
✄
✆
2
AB ☎ 2
✠ , i.e.,
XB 1
✝
✟
✡
AB 2
XB 1
✞
or,
or,
or,
XB
AB = 1 2
XB
1– AB
=
1–
AB – XB 2 ☛ 1
☞ , i.e.,
AB 2
1
2
AX 2 1 ☛
= 2 2 ✌
.
☞
AB 2 2
EXERCISE 6.4
1. Let ✍ ABC ~ ✍ DEF and their areas be, respectively, 64 cm 2 and 121 cm 2 . If EF = 15.4
cm, find BC.
2. Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O.
If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.