Question

In the given, PT touches the circle with centre Q at point R. Diameter SQ is product to meet the tangent TR at P. $\angle SPR=x^{\circ }$ and $\angle QRP=y^{\circ }$
Which one of the following statements are true?

• A. $x^{\circ }+2y^{\circ }={90}^{\circ }$
• B. $x^{\circ }+2y^{\circ }={50}^{\circ }$
• C. $x^{\circ }+2y^{\circ }={80}^{\circ }$
• D. $x^{\circ }+2y^{\circ }={190}^{\circ }$

Answer 1

The correct option is A

$x^{\circ }+2y^{\circ }={90}^{\circ }$

Given, PT touches the circle with centre Q at point R. Diameter SQ is product to meet the tangent TR at P.

Given $\angle SPR=x^{\circ }$ and $\angle QRP=y^{\circ }$

$\angle QRP=\angle OSR=y$

(Angles in the alternate segment)

But OS = OR (radii of the same circle)

$\therefore \angle ORS=\angle OSR=y^{\circ }$

$\therefore OQ=OR$ (radii of the same circle)

$\therefore \angle OQR=\angle ORQ={90}^{\circ }-y^{\circ }$ .....(i)

$(OR\perp PT)$

But in $\Delta PQR$,

Ext. $\angle OQR=x^{\circ }+y^{\circ }$ ....(ii)

from (i) and (ii)

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

$\Rightarrow x^{\circ }+2y^{\circ }={90}^{\circ }$