An isosceles triangle $PQR$, where $PQ=PR$, is inscribed in a circle with centre $O$ and radius $1$

Question

An isosceles triangle $PQR$, where $PQ=PR$, is inscribed in a circle with centre $O$ and radius $1$ unit. It is given that the angle $QOR=4	heta $ radians, where $	heta $ is an acute angle. Show that $A$, the area of triangle $PQR$, can be expressed as $\left( \sin 2	heta ight)\left( 1+\cos 2	heta ight)$. [3] Hence use differentiation to prove that $	heta $ varies, $A$ is a maximum when triangle $PQR$ is an equilateral triangle. [5]

Accepted Answer

(i) $A=\sin2\theta(1+\cos2\theta)$; (ii) maximum at $\theta=\frac\pi6$, and $\triangle PQR$ is equilateral

2022 ACJC J2 MYE Q3

2022 ACJC J2 MYE Q3

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