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13. In a triangleABC AD perpendicular BC. Prove that AB² + CD²= AC² + DB²​

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In triangle ABC, AD perpendicular to BC and AD2 =BD * CD.prove that triangle ABC is a right triangle

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asked Oct 30, 2019 in Important Questions by megha00 (-7,831 points)

In triangle ABC, AD perpendicular to BC and AD2 =BD * CD.prove that triangle ABC is a right triangle

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answered Oct 30, 2019 by manish56 (-1,810 points)

Given : In triangle ABC , AD is perpendicular to BC and AD² = BD.DC

To prove : BAC = 90°

Proof : in right triangles ∆ADB and ∆ADC

So, Pythagoras theorem should be apply ,

Then we have ,

AB² = AD² + BD² ----------(1)

AC²= AD²+ DC² ---------(2)

AB² + AC² = 2AD² + BD²+ DC²

= 2BD . CD + BD² + CD² [ ∵ given AD² = BD.CD ]

= (BD + CD )² = BC²

Thus in triangle ABC we have , AB² + AC²= BC²

hence triangle ABC is a right triangle right angled at A

∠ BAC = 90°