Question : The ratio of the area of two isosceles triangles having the same vertical angle (i.e. the angle between equal sides) is 1 : 4. The ratio of their heights is:
Option 1: 1 : 4
Option 2: 2 : 5
Option 3: 1 : 2
Option 4: 3 : 4
Correct Answer: 1 : 2
Solution :
$\angle$ BAC = $\angle$ QPR
$\angle$ ABC = $\angle$ ACB and $\angle$ PQR = $\angle$ PRQ
$\angle$ ABC = $\angle$ ACB = $\angle$ PQR = $\angle$ PRQ
$\triangle$ BAC ~ $\triangle$ QPR
$\frac{area(\triangle BAC)}{area(\triangle PQR)}$ = $\frac{h^2}{H^2}$
⇒ $\frac{1}{4}$ = $\frac{h^2}{H^2}$
⇒ $\frac{h}{H}$ = $\frac{1}{2}$
Hence, the correct answer is 1 : 2.
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