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 isA. 73°B. 64C. 53°D. 44°

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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