In right-angled triangle \( A B C \) in which \( \angle C=90 \), if \( D \) is the mid-point of \( B C \) prove that \( A B^{2}=4 A D^{2}-3 A C^{2} \).
Given:
In right-angled triangle \( A B C \), \( \angle C=90 \) and \( D \) is the mid-point of \( B C \).
To do:
We have to prove that \( A B^{2}=4 A D^{2}-3 A C^{2} \).
Solution:
\( D \) is the mid-point of \( B C \). Therefore, $BC=2CD=2BD$.
In $\triangle ABC$, by Pythagoras theorem,
$AB^2=AC^2+BC^2$
$AB^2=AC^2+(2CD)^2$ ($BC=2CD$)
$AB^2=AC^2+4CD^2$.....(i)
In $\triangle ACD$, by Pythagoras theorem,
$AD^2=AC^2+CD^2$
$CD^2=AD^2-AC^2$....(ii)
Substituting equation (ii) in equation (i), we get,
$AB^2=AC^2+4(AD^2-AC^2)$
$AB^2=AC^2+4AD^2-4AC^2$
$AB^2=4AD^2-3AC^2$
Hence proved.
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