Question : PA and PB are two tangents from a point P outside the circle with centre O at the points A and B on it. If $∠ A P B = 130^{\circ}$ , then $∠ O A B$ is equal to:
Option 1: 45°
Option 2: 50°
Option 3: 35°
Option 4: 65°

Question

Question : PA and PB are two tangents from a point P outside the circle with centre O. If A and B are points on the circle such that $∠ A P B = 100^{\circ}$ , then $∠ O A B$ is equal to:
Option 1: $45^{\circ}$
Option 2: $35^{\circ}$
Option 3: $70^{\circ}$
Option 4: $50^{\circ}$

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Answer 1

Correct Answer: 65°

Solution :
Given: PA and PB are tangents
$∠ O A P = 90^{\circ}$
$∠ O B P = 90^{\circ}$
As, OAPB is a quadrilateral
$∠ O A P + ∠ A P B + ∠ P B O + ∠ B O A = 360^{\circ}$
⇒ $90^{\circ} + 130^{\circ} + 90^{\circ} + ∠ B O A = 360^{\circ}$
⇒ $∠ B O A = 360^{\circ} - 308^{\circ}$
⇒ $∠ B O A = 50^{\circ}$
As, OA = OB (Radius)
In ΔOAB, $∠ O A B = ∠ O B A$ (In a triangle angles opposite to equal sides are equal)
$∠ O A B + ∠ O B A + ∠ B O A = 180^{\circ}$
⇒ $2 \times ∠ O A B + 50^{\circ} = 180^{\circ}$
⇒ $∠ O A B = \frac{(180^{\circ} - 50^{\circ})}{2}$
∴ $∠ O A B = 65^{\circ}$
Hence, the correct answer is $65^{\circ}$ .

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Question : PA and PB are two tangents from a point P outside the circle with centre O at the points A and B on it. If ∠APB=130∘, then ∠OAB is equal to:Option 1: 45°Option 2: 50°Option 3: 35°Option 4: 65°

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Question : PA and PB are two tangents from a point P outside the circle with centre O at the points A and B on it. If $∠ A P B = 130^{\circ}$ , then $∠ O A B$ is equal to:
Option 1: 45°
Option 2: 50°
Option 3: 35°
Option 4: 65°