Hi, I have a question and I hope anyone could answer it:

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

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

Hi, I have a question and I hope anyone could answer it:

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

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Answer:

Given that in the question, triangle ABC, AD is perpendicular to BC and AD² = BD.DC

To prove : BAC = 90°

in right triangles ∆ADB and ∆ADC

So, Pythagoras theorem should be applied

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°