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# f4math 難題(quickly)

PA and QC are tangents.show that PQ=PA

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- 魏王將張遼Lv 71 decade agoFavorite Answer
Pls. check if the question needs the proof to be PQ = QA instead of PQ = PA.

Join AC and let ∠ABC = θ. Then ∠BCA = 90 (Angle in semi-circle)

∠QAC = ∠QCA = θ (Angle in alternate segment)

So QA= QC since QCA is an isos. triangle.

∠PCQ = 180 - ∠BCA - ∠QCA = 90 - θ

Also ∠CPQ = 180 - ∠BAP - ∠PBA = 90 - θ as ∠BAP = 90 for the reason of diameter perpendicular to tangent.

So with ∠CPQ = ∠PCQ, CPQ is an isos. triangle with PQ = CQ

Finally, PQ = CQ = QA

Source(s): My Maths knowledge- Login to reply the answers

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