Q. 275.0( 3 Votes )

In the given figure, PQ is a tangent to a circle with center O. A is the point of contact. If PAB = 67°, then the measure of AQB is


A. 73°

B. 64

C. 53°

D. 44°

Answer :


In the given Figure, Join OA


Now,


OA PQ


[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]


OAP = OAQ = 90° [1]


OAB + PAB = 90°


OAB + 67° = 90°


OAB = 23°


Now,


BAC = 90°


[Angle in a semicircle is a right angle]


OAB + OAC = 90°


23° + OAC = 90°


OAC = 67°


OAQ = 90° [From 1]


OAC + CAQ = 90°


67° + CAQ = 90°


CAQ = 23° [2]


Now,


OA = OC


[radii of same circle]


OCA = OAC


[Angles opposite to equal sides are equal]


OCA = 67°


OCA + ACQ = 180° [Linear Pair]


67° + ACQ = 180°


ACQ = 113° [3]


Now, In ACQ By Angle Sum Property of triangle


ACQ + CAQ + AQC = 180°


113° + 23° + AQC = 180° [By 2 and 3]


AQC = 44°

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