Question

In the given figure, $O$ 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

Answer: (C)

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

In $△OQROR=OQ=$ radius

$\Rightarrow \angle OQR=\angle ORQ=y^o$

Thus by angle sum property, we have

$\angle OQR+\angle ORQ+\angle QOR={180}^{o}2\angle ORQ+\angle QOR={180}^o\angle QOR=180-2\angle ORQ=180-2x^o$

Also, $\angle QOR=2\angle QPR$ (half angle property)

$180-2x=2y180=2x+2y{x}^o+y^o={90}^o$