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In a right angled triangle if …

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In a right angled triangle if a perpendicular is drawn from the right angle to the hypotenuse then prove that the area of the square of the perpendicular is equal to the area of the rectangle formed by the two segments of the hypotenuse

    • Posted by Keshav Goyal 6 years, 5 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Sia ? 6 years, 5 months ago

    GIVEN A right triangle ABC right-angled at A, {tex}A D \perp B C.{/tex}
    TO PROVE AD2 = BD {tex}\times{/tex} CD

     Since {tex}\Delta{/tex} ABD and {tex}\Delta{/tex}ACD are right triangles.
    {tex}\therefore{/tex} AB2 = AD2 + BD2 ......(i)
    and, AC2 = AD2 + CD2 .......(ii)                              
    Adding equations (i) and (ii), we get
    AB2 + AC2 = 2AD2 + BD2 + CD2
    {tex}\Rightarrow{/tex} BC2 = 2 AD2 + BD2 + CD2 [{tex}\because{/tex} {tex}\Delta{/tex}ABC is right-angled at A {tex}\therefore{/tex} AB2 + AC2 = BC2]
    {tex}\Rightarrow{/tex} (BD + CD)2 = 2AD2 + BD2 + CD2
    {tex}\Rightarrow{/tex} BD2 + CD2 + 2 BD {tex}\times{/tex} CD = 2 AD2 + BD2 + CD2
    {tex}\Rightarrow{/tex} 2BD {tex}\times{/tex} CD = 2 AD2
    {tex}\Rightarrow{/tex} AD2 = BD {tex}\times{/tex} CD
    {tex}\Rightarrow{/tex} Hence, AD2 = BD {tex}\times{/tex} CD.

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