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Tangent Perpendicular to Radius at Point of Contact

In the figure...

Question

$In the figure given below, O is the center of the circle, CA is tangent at A and CB is tangent at B drawn to the circle.$

$If ∠ A C B = 75^{\circ}, then ∠ A O B =$ ______.

$75^{\circ}$

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$85^{\circ}$

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$95^{\circ}$

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$105^{\circ}$

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Solution

The correct option is D
$105^{\circ}$

$∠ O A C = ∠ O B C = 90^{\circ}$
$(∵ The tangent at any point on a circle$
$is perpendicular to the radius through$ $that point.)$

$∠ O A C + ∠ O B C + ∠ A C B + ∠ A O B = 360^{\circ}$
$(∵ Sum of angles of a quadrilateral)$

$90^{\circ} + 90^{\circ} + 75^{\circ} + ∠ A O B = 360^{\circ}$

$∠ A O B = 360^{\circ} - 255^{\circ} = 105^{\circ}$
$∠ A O B = 105^{\circ}$

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Similar questions

$In the figure given below, O is the center of the circle, CA is tangent at A and CB is tangent at B drawn to the circle.$

$If ∠ A C B = 75^{\circ}, then ∠ A O B =$ ______.

Q. In figure,

O

is the centre of the circle,

C A

is tangent at

A

and

C B

is tangent at

B

drawn to the circle. If

∠ A C B = 75^{o}

∠ A O B =

Q. In the given figure, O is the center of a circle and two tangents CA, CB are drawn on the circle from a point C lying outside the circle. Prove that

∠ A O B

and

∠ A C B

are supplementary.

Q. In the given figure AB is a chord of a circle with centre O and BT is a tangent. If

∠ A C B = 75^{\circ}

, then

∠ A B T =

___

___.

In the given figure, $O$ is the centre of a circle. If $∠ A O B = 100^{\circ}$ and $∠ A O C = 90^{\circ}$ , then $∠ B A C = ?$

(a) $85^{\circ}$

(b) $80^{\circ}$

(d) $75^{\circ}$

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In the figure given below, O is the center of the circle, CA istangent at A and CB is tangent at B drawn to the circle. If ∠ ACB=75∘, then ∠AOB= ______.

$In the figure given below, O is the center of the circle, CA is tangent at A and CB is tangent at B drawn to the circle.$

$If ∠ A C B = 75^{\circ}, then ∠ A O B =$ ______.