Point P lies in the exterior of circle with centre O. Tangents through P touch the circle at A and B. If the angle formed by PA and PB is of 80°, then find angle POA.
Answers
The measure of angle POA is 50°
- Given,
P is an external point and PA and PB are tangents to circle with center O such that they meet the circle at A and B.
∠APB = 80°
- Consider triangles, OAP and OBP
OA = OB (radii of circle)
OP = OP (common side)
PA = PB (lengths of tangents from a common point)
∴ OAP≅OBP [from SSS congruence]
⇒ ∠OPA = OPB (By CPCT)
⇒ ∠OPA = OPB = 40°
Also
∠OAP = OBP = 90° [∵ tangent is perpendicular to radius]
Now in ΔOAP
∠A + ∠O + ∠P = 180° [angle sum property]
90° + ∠O + 40° = 180°
∠O = 180° - 130°
∴ ∠POA = 50°
