AP and AQ are two tangents drawn to a circle with center O from an external point A. If ∠PAQ = 40°, then find the measure of ∠POQ.

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Accepted Answer

Given: AP and AQ are tangents to the circle from external point A, and ∠PAQ = 40°.

Step 1: Recall that the tangents from an external point to a circle are equal in length and the angle between the tangents is ∠PAQ.

Step 2: The angle subtended by the chord POQ at the center O is related to the angle between the tangents at A by the formula:

∠POQ = 2 × (180° - ∠PAQ)

Step 3: Substitute the given value:

∠POQ = 2 × (180° - 40°) = 2 × 140° = 280°

This seems incorrect because the angle at the center cannot be more than 360°, and the angle between the tangents is related differently.

Step 4: Actually, the correct relationship is:

The angle between the two tangents (∠PAQ) is supplementary to the angle subtended by the chord POQ at the center O.

So,

∠POQ = 360° - 2 × ∠PAQ

Step 5: Calculate:

∠POQ = 360° - 2 × 40° = 360° - 80° = 280°

This again seems off because angle POQ is an angle at the center between points P and Q on the circle.

Step 6: The correct formula is:

The angle between the tangents (∠PAQ) = 180° - ∠POQ/2

Therefore,

∠PAQ = 180° - (∠POQ / 2)

Step 7: Rearranging to find ∠POQ:

∠POQ / 2 = 180° - ∠PAQ

∠POQ = 2 × (180° - ∠PAQ) = 2 × (180° - 40°) = 2 × 140° = 280°

This again gives 280°, which is not possible for the angle at the center between two points on the circle.

Step 8: Actually, the angle between the tangents is equal to the difference between 180° and the angle subtended by the chord at the center:

∠PAQ = 180° - ∠POQ

Therefore,

∠POQ = 180° - ∠PAQ = 180° - 40° = 140°

Hence, the angle ∠POQ is 140°.

Final Answer: 140°

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