Question

BL and CM are medians of ABC right angled at A. Prove that BL2 + CM2 = BC2 + LM2
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Answer 1

$\textbf{Given:}$

$\text{BL and CM are medians of right triangle ABC}$

$\textbf{To prove:}$

$BL^2+CM^2=BC^2+LM^2$

$\textbf{Solution:}$

$\text{In \triangle BAL, by pythagoras theorem}$

$\bf\,BL^2=AB^2+AL^2$.......(1)

$\text{In \triangle MAC, by pythagoras theorem}$

$\bf\,CM^2=AM^2+AC^2$.......(2)

$\text{In \triangle BAC, by pythagoras theorem}$

$\bf\,BC^2=AB^2+AC^2$.......(3)

$\text{In \triangle MAL, by pythagoras theorem}$

$\bf\,ML^2=AM^2+AL^2$.......(4)

$\text{Adding (1) and (2), we get}$

$BL^2+CM^2=(AB^2+AL^2)+(AM^2+AC^2)$

$BL^2+CM^2=(AB^2+AC^2)+(AL^2+AM^2)$

$\text{Using (3) and (4) we get}$

$BL^2+CM^2=BC^2+ML^2$

$\implies\boxed{\bf\,BL^2+CM^2=BC^2+LM^2}$