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Questions
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 ______.
Options
50°
60°
70°
80°
40°
Solution
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.
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