PA and QC are tangents.show that PQ=PA
- 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 = QASource(s): My Maths knowledge