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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Question : ABC is an isosceles triangle and $\overline{AB} = \overline{AC} = 2a$ units and $\overline{BC} = a$ unit. Draw $\overline{AD} \perp \overline{BC}$, then find the length of $\overline{AD}$.
Question : $\triangle PQR$ is right-angled at $Q$. The length of $PQ$ is 5 cm and $\angle P R Q=30^{\circ}$. Determine the length of the side $QR$.
Question : A circle is inscribed in $\triangle $PQR touching the sides QR, PR and PQ at the points S, U and T, respectively. PQ = (QR + 5) cm, PQ = (PR + 2) cm. If the perimeter of $\triangle $PQR is 32 cm, then PR is equal to:
Question : In $\triangle$PQR, a straight line parallel to the base, QR cuts PQ at X and PR at Y. If PX : XQ = 5 : 6, then XY : QR will be:
Question : In an isosceles triangle, the length of each equal side is twice the length of the third side. The ratios of areas of the isosceles triangle and an equilateral triangle with the same perimeter are:
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