Question

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

If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other atan angle of 80° as shown in the given figure, then the measure of ∠POA is ______.

पर्याय

• 50° 

• 60°

• 70° 

• 80°

• 40°

Answer 1

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

Explanation:

It is given that 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 = 90º

∠OAP = 90º

In AOBP,

Sum of all interior angles = 360°

∠OAP + ∠APB + ∠PBO + ∠BOA = 360°

⇒ 90° + 80° + 90° + ∠BOA = 360°

⇒ 260° + ∠BOA = 360°

⇒ ∠BOA = 360° – 260°

⇒ ∠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)

A ↔ B, P ↔ P, O ↔ O

And thus, ∠POB = ∠POA

`angle"POA" = 1/2 angle"AOB" = (100^@)/2 = 50^@`

Hence, alternative 50° is correct.