Let C be a circle with centre O and P be an external point to C. Let PA and PB…
2024
Let C be a circle with centre O and P be an external point to C. Let PA and PB be two tangents to C with A and B being the points of tangency, respectively. If PA and PB are inclined to each other at an angle of 60°, then find ∠POA.
- A.
40°
- B.
60°
- C.
80°
- D.
30°
Show answer & explanation
Correct answer: B
Properties of Tangents:
Tangents from an external point (P) to a circle are equal in length (PA = PB).
The line segment (PO) connecting the external point to the center of the circle bisects the angle between the tangents.
The radius (OA) drawn to the point of tangency (A) is perpendicular to the tangent (PA). Therefore, angle OAP = 90°.
Step-by-Step Calculation:
Find angle APO: Since PO bisects the angle between the tangents (60°), angle APO = 60° / 2 = 30°.
Analyze triangle OAP: This is a right-angled triangle where angle OAP = 90° and angle APO = 30°.
Calculate angle POA: The sum of angles in a triangle is 180°.
Angle POA = 180° - (Angle OAP + Angle APO)
Angle POA = 180° - (90° + 30°)
Angle POA = 180° - 120° = 60°