Question

The triangle PQR is inscribed in the circle $x^2+y^2=25.$ If $Q$ and $R$ have coordinates $(3,4)$ and $(-4,3),$ respectively, then $\angle QPR$ is equal to

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is C $\frac{\pi }{4}$

Now, length of chord $QR=\sqrt{(-4-3{)}^2+(3-4{)}^2}=5\sqrt{2}$

Also, $OR=OQ=5$ (radii of the circle)

$\therefore \angle ROQ={90}^{\circ }$

Using the theorem that arc subtended by a chord is twice the angle subtended by the chord at the arc of the circle,

we have $\angle QPR={45}^{\circ }=\frac{\pi }{4}$