Question

In the figure below, O is the centre of the circle and $\angle QPR=x^{\circ };\angle ORQ=y^{\circ }$. Find $x^{\circ }+y^{\circ }$.

• A. $x^{\circ }+y^{\circ }={120}^{\circ }$
• B. $x^{\circ }+y^{\circ }={180}^{\circ }$
• C. $x^{\circ }+y^{\circ }={90}^{\circ }$
• D. $x^{\circ }+y^{\circ }={150}^{\circ }$

Answer 1

The correct option is C

$x^{\circ }+y^{\circ }={90}^{\circ }$

The angle subtended by an arc of a circle at the centre of the circle is twice the angle subtended by it at any other point on the circle.
So, $\angle QOR=2\angle QPR=2x^{\circ }$
$OQ=OR$ ($\because$ Radius of the same circle)
So, $\angle OQR=\angle ORQ$ ($\because$ Angle opposite to equal sides are equal)

In $\Delta OQR$
$2x^{\circ }+y^{\circ }+y^{\circ }={180}^{\circ }$
$2x^{\circ }+2y^{\circ }={180}^{\circ }$
$\therefore x^{\circ }+y^{\circ }={90}^{\circ }$