In a right $ \triangle A B C $ right-angled at $ C $, if $ D $ is the mid-point of $ B C $, prove that $B C^{2}=4(A D^{2}-A C^{2})$.

Given:

In a right \( \triangle A B C \) right-angled at \( C \), \( D \) is the mid-point of \( B C \).
To do:

We have to prove that $B C^{2}=4(A D^{2}-A C^{2})$.

Solution:

In $\triangle ADC$, by Pythagoras theorem,

$AD^2=AC^2+DC^2$

$DC^2=AD^2-AC^2$.....(i)

$BC=2DC$   (\( D \) is the mid-point of \( B C \))

$BC^2=(2DC)^2$

$BC^2=4DC^2$

$BC^2=4(AD^2-AC^2)$    (From (i))

Hence proved.

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Updated on: 10-Oct-2022

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