Question:
$P, Q, S, R$ are points on a circle of radius $r$, such that $PQR$ is an equilateral triangle and $PS$ is a diameter. What is the perimeter of quadrilateral $PQSR$?
$P, Q, S, R$ are points on a circle of radius $r$, such that $PQR$ is an equilateral triangle and $PS$ is a diameter. What is the perimeter of quadrilateral $PQSR$?
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In circle problems, use chord length formula $PQ = 2r\sin(\theta/2)$.
Updated On: Jul 31, 2025
- $2r(1+\sqrt{3})$
- $2r(2+\sqrt{3})$
- $r(1+\sqrt{5})$
- $2r+\sqrt{3}$
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The Correct Option is A
Solution and Explanation
In the circle, $PQR$ equilateral implies chord $PQ = QR = r\sqrt{3}$. $PS$ is diameter = $2r$, $SR = r\sqrt{3}$. Perimeter = $PQ + QR + RS + SP = r\sqrt{3} + r\sqrt{3} + r\sqrt{3} + 2r = 2r(1+\sqrt{3})$.
\[
\boxed{2r(1+\sqrt{3})}
\]
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