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Question - Pythagoras Theorem and Its Converse

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Question

In a right triangle ABC right-angled at C, P and Q are the points on the sides CA and CB respectively, which divide these sides in the ratio 2 : 1. Prove that

`(i) 9 AQ^2 = 9 AC^2 + 4 BC^2`

`(ii) 9 BP^2 = 9 BC^2 + 4 AC^2`

`(iii) 9 (AQ^2 + BP^2 ) = 13 AB^2`

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