If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to

Question

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80&#xB0;, then &#x2220; POA is equal to A. 50&#xB0; B. 60&#xB0; C. 70&#xB0; D. 80&#xB0;

Philoid · Accepted Answer

(A)

Given,

PA and PB are tangents.

Therefore, the radius drawn to these tangents will be perpendicular to the tangents.

Thus, OA ⊥ PA and OB ⊥ PB

∠OBP = 90°

∠OAP = 90°

In quadrilateral AOBP,

Sum of all interior angles = 360°

∠OAP + ∠APB +∠PBO + ∠BOA = 360° 90° + 80° +90° + ∠BOA = 360°

∠BOA = 100°

In ΔOPB and ΔOPA,

AP = BP (Tangents from a point)

OA = OB (Radii of the circle)

OP = OP (Common side)

Therefore, ΔOPB ≅ ΔOPA (SSS congruence criterion)

And thus, ∠POB = ∠POA

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal toA. 50° B. 60°C. 70° D. 80°

If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
A. 50° B. 60°

C. 70° D. 80°