Find the ratio of the area of triangle XBY to the area of triangle ABC for the given measurements.

Question

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Find the ratio of the area of triangle XBY to the area of triangle ABC for the given measurements, if

XY = 2, AC = 3


A. 1/3
B. 2/3
C. 4/9

Answer 1

Let sides of traingle ABC be a,b,c
Since triangle ABC and triangle XBY are similar
 
XY/AC = YB/CB = XB/AB = 2/3
 
Therefore sides of triangle XBY will be 2/3a, 2/3b, 2/3c
 
Use Heron's formula to find area of a triangle
 
Area of triangle ABC = sqrt ( s(s-a)(s-b)(s-c)) where s = 1/2(a+b+c)
 
Area of triangle XBY = sqrt( s'(s'-a')(s'-b')(s'-c') where s' = 1/2(2/3a + 2/3b + 2/3c) = 2/3s, a'=2/3a, b'=2/3b, c'=2/3c
 
Therefore area of traingle XBY = sqrt ( 2/3s(2/3s - 2/3a)(2/3s - 2/3b)(2/3s - 2/3c)
 
= sqrt(2/3 x 2/3 x 2/3x 2/3 (s(s-a)(s-b)(s-c)))
= 4/9 sqrt(s(s-a)s-b)(s-c)) = 4/9 Area of triangle ABC
 
Therefore ratio of the area of triangle XBY to the area of triangle ABC = 4/9