Points A and B are on a circle with centre O. PAM and PBN are tangentsto the circle at A and B respectively from a point P outside the circle. Point Q is on the major arc AB such that ∠QAM = 58° and ∠QBN = 50°, then find the measure (in degrees) of ∠APB.
Answer & explanation
Correct answer: option 1
The angles of an alternate segment subtended by a tangent to a circle are equal.
\(\angle\)QAM = \(\angle\)ABQ = \({58}^\circ\)
\(\angle\)QBN = \(\angle\)BAQ = \({50}^\circ\)
So, \(\angle\) = \({72}^\circ\)
\(\angle\)BOA = 2\(\angle\)AQB = 2 x \({72}^\circ\) = \({144}^\circ\)
In quadrilateral PBOA,
\(\angle\)APB + \(\angle\)BOA = \({180}^\circ\) (because OB is perpendicular to PN and PA is perpendicular to PM )
= \(\angle\)APB = 180 -144
= \(\angle\)APB = \({36}^\circ\)
Therefore, \(\angle\)APB is \({36}^\circ\).