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PAQ is a tangent to the circle with center O at a point A as shown in figure. If $∠ OBA = 35 °,$ find the value of $∠ BAQ$ and $∠ ACB$

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30 o

90 o

55 o

60 o

answer is B.

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Detailed Solution

Given that PAQ is a tangent to the circle with center O at a point A.
We have to find

∠ BAQ

and

∠ ACB .

Tangent of a circle at any point in a circle is perpendicular to the radius through the point of contact.
From the figure,
Question Image

In Δ O A B, O A = O B {∵ radii of a circle ∠ O B A = ∠ O A B {∵ Opposite angle of the equal side}

It is given that,

∠ O B A = 35 ° .

So,

∠ O A B = 35 °

.
From the figure, PAQ is the tangent at point A. So, the tangent of a circle at any point in a circle is perpendicular to the radius through the point of contact. That means,

O A ⊥ P Q,

and

∠ O A Q = ∠ O A P = 90 °

∠ O A Q = ∠ O A B + ∠ B A Q ⇒ 90 ° = 35 ° + ∠ B A Q ⇒ ∠ B A Q = 90 ° − 35 ° ⇒ ∠ B A Q = 55 °

Due to the angle made in the alternate segment,

∠ B A Q = ∠ A C B = 55 °

The values are,

∠ B A Q = 55 °,

and

∠ A C B = 55 °

Therefore, the correct option is 2.

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