Question

In the figure below, O is the centre of the circle and $\angle QPR=x^{\circ },\angle ORQ=y^{\circ }.$ Which statement is true about $x^{\circ }$ and $y^{\circ }$?

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

Angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by it at any other point on the circle.
So, $\angle QOR=2\angle QPR=2x.$
Note that $OQ=OR$ ($\because$ radius of the same circle)
$⟹\angle OQR=\angle ORQ$ ($\because$ angles opposite to equal sides are equal)
In $\Delta OQR,$
$2x^{\circ }+y^{\circ }+y^{\circ }={180}^{\circ }$
i.e., $2x^{\circ }+2y^{\circ }={180}^{\circ }$
$\therefore x^{\circ }+y^{\circ }={90}^{\circ }$