$Δ A B C$ is a right-angled triangle right angled at $B .$ $B D$ is perpendicular to $A C .$ If $A D = 8 c m$ and $D C = 2 c m,$ then find the length of $B D .$

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Solution

In $△ A B C, A B^{2} + B C^{2} = 100 \dots (1)$
In $△ A D B, A D^{2} + B D^{2} = A B^{2} \dots (2)$
In $△ B D C, B D^{2} + D C^{2} = B C^{2} \dots (3)$
Adding $(2), (3)$
$⟹ 2 B D^{2} + 8^{2} + 2^{2} = 100$
$⟹ B D = 4$

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