Question : $\triangle PQR$ is an isosceles triangle and $PQ=PR=2a$ unit, $QR=a$ unit. Draw $PX \perp QR$, and find the length of $PX$.
Option 1: $\sqrt{5} a$
Option 2: $\frac{\sqrt{5} a}{2}$
Option 3: $\frac{\sqrt{10} a}{2}$
Option 4: $\frac{\sqrt{15} a}{2}$
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Correct Answer: $\frac{\sqrt{15} a}{2}$
Solution :
Given: $PQ = PR = 2a$ and $QR=a$
⇒ $PX = \sqrt{(PQ^2 - (\frac{QR}{2})^2)}$
= $\sqrt{(4a^2 - \frac{a^2}{4})}$
= $\frac{\sqrt{15}a}{2}$
Hence, the correct answer is $\frac{\sqrt{15}a}{2}$.
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