Question

In the adjoining figure, O is the centre of the circle and PQ, RS are its equal chords, OD $\perp$ PQ and OE $\perp$ RS. If $\angle DOE={130}^{\circ },$ then $\angle PDE$ = ___.

• A. ${50}^{\circ }$
• B. ${65}^{\circ }$
• C. ${40}^{\circ }$
• D. ${70}^{\circ }$

Answer 1

The correct option is B

${65}^{\circ }$

Since equal chords are equidistant from the centre, we have OD = OE.

$⟹\angle ODE=\angle OED=x\left(\text{say}\right)$

$\therefore x+x+130={180}^{\circ }\left(\text{Angle sum property}\right)$

$⟹2x=50⟹x={25}^{\circ }$

i.e., $\angle ODE={25}^{\circ }$

$OD\perp PQ⟹\angle PDO={90}^{\circ }$

$\therefore \angle PDE=\angle PDO-\angle ODE={90}^{\circ }-{25}^{\circ }={65}^{\circ }$

Thus, $\angle PDE={65}^{\circ }.$