Question

$△PQR$ is inscribed in the circle $x^2+y^2=25$, Q = (3,4) and R = (-4,3). The measure of $\angle QPR$ (in radius)

• A. $\frac{\pi }{6}or\frac{5\pi }{6}$
• B. $\frac{\pi }{3}or\frac{2\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}or\frac{3\pi }{4}$

Answer 1

The correct option is D $\frac{\pi }{4}or\frac{3\pi }{4}$

Distance between the 2 points can be found to be $\sqrt{50}$ units.

Construct a perpendicular on the chord from the centre. This must bisect the chord giving half length as $\frac{5}{\sqrt{2}}$ units.The hypotenuse is the radius of 5 units

Hence $\cos \theta =\frac{1}{\sqrt{2}}$

So $\theta =45degrees$

Hence $\angle POR={45}^{\circ }$

$\angle QOR=2\angle POR={90}^{\circ }$

So the chord will subtend half the angle on the circle for major arc while it will subtend $\frac{1}{2}({360}^{\circ }-{90}^{\circ })$ on the minor arc.

Hence answer is option D.