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In figure if PQR is the tangent to a circle at Q whose center is O, AB is a chord parallel to PR and $∠ BQR=70 o,$ then $∠ AQB$ is equal to

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45 °

35 °

20 °

40 °

answer is B.

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Detailed Solution

Given that PQR is the tangent to a circle at Q whose center is O, AB is a chord parallel to PR and

∠ BQR=70 o .

We have to find

∠ A Q B .

The sum of all the three angles of a triangle is equal to

180 o

From figure

O D ⊥ A B

and O D bisects A B As angles opposite to the equal sides of a triangle are equal,

∠ A B Q = ∠ B Q R ⇒ ∠ A B Q = ∠ B Q R = 70 ° ⇒ Also, ∠ O Q P = ∠ O Q R = 90 ° .

Since the angles and sides of

Δ A D Q

and

Δ B D Q

are similar, both the triangles are congruent. So,

∠ Q A D = ∠ Q B D = ∠ A B Q = 70 From △ A B Q, ∠ A + ∠ B + ∠ A Q B = 180 ° ⇒ 70 ° + 70 ° + ∠ A Q B = 180 ° ⇒ ∠ A Q B = 180 ° − 140 ° ⇒ ∠ A Q B = 40 °

Therefore, the correct option is 2.

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