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In the given figure, A and B are the centers of two circles. DC is a common tangent. If $∠ DPM = 40 °$ ,then find $∠ RQC .$ .

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30 o

90 o

50 o

60 o

answer is B.

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Detailed Solution

Given that A and B are the centers of two circles. DC is a common tangent.
We have to find the value of

∠ RQC .

The angle subtended by an arc at the center is double the angle subtended at the circumference.
The required figure geometry is shown below,
Question Image

In the figure, Since DM arc makes an angle

40 o

with the circumference. Then, the angle subtended by an arc at the center is double the angle subtended at the circumference. So,

∠ D A M = 2 × ∠ D P M ⇒ ∠ D A M = 2 × 40 ° ⇒ ∠ D A M = 80 °

In the quadrilateral ABCD by using the sum of the angle property,

80 ° + 90 ° + 90 ° + x = 360 ° ⇒ x = 180 ° − 80 ° ⇒ x = 100 °

Since the angle subtended by an arc at the center is double the angle subtended at the circumference. So, in the arc CR

∠ R Q C = 12 × ∠ R B C ⇒ ∠ R Q C = 12 × 100 ° ⇒ ∠ R Q C = 50 °

The value of

∠ R O C

50 ° .

Therefore, the correct option is 2.

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