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

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  1. 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°

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