Practicing Success
Point A and B are on a circle with centre O. PA and PB are tangents to the circle from an external point P. If PA and PB are inclined to each other at 42°, then find the measure of ∠OAB. |
42° 21° 69° 25° |
21° |
Consider the quadrilateral AOBP, ⇒ \(\angle\)A + \(\angle\)O + \(\angle\)B + \(\angle\)P = \({360}^\circ\) ⇒ \({90}^\circ\) + \(\angle\)O + \({90}^\circ\) + \({42}^\circ\) = \({360}^\circ\) ⇒ \(\angle\)O = \({360}^\circ\) - \({90}^\circ\) - \({90}^\circ\) - \({42}^\circ\) ⇒ \(\angle\)O = \({138}^\circ\) ⇒ \(\Delta \)OAB is an isosceles triangle since OA and OB are the radii of the circle. So, \(\angle\)OAB = \(\angle\)OBA = \({R}^\circ\) (let) ⇒ 2\({R}^\circ\) + \({138}^\circ\) = \({180}^\circ\) ⇒ 2\({R}^\circ\) = \({42}^\circ\) ⇒ R = \({21}^\circ\) ⇒ \(\angle\)OAB = \({21}^\circ\) Therefore, \(\angle\)OAB is \({21}^\circ\). |