In a right △ABC, a perpendicular BD is drawn on to the largest side from the

Question

In a right $△$ ABC, a perpendicular BD is drawn on to the largest side from the opposite vertex. Which of the following does not give the ratio of the areas of $△$ ABD and $△$ ACB?

Options:

A .

({\frac{A B}{A C})}^{2}

B .

({\frac{A D}{A B})}^{2}

C .

({\frac{B D}{C B})}^{2}

D .

({\frac{A B}{A D})}^{2}

Answer: Option D
:
D

Consider ΔABD and ΔACB:

∠BAD = ∠BAC [common angle]

∠BDA = ∠ABC [ $90^{\circ}$ ]

By AA similarity criterion,
$△ A B D ~ △$ ACB

Hence,

$\frac{a r (Δ A B D)}{a r (Δ A C B)} = ({\frac{A B}{A C})}^{2} = ({\frac{A D}{A B})}^{2} = ({\frac{B D}{C B})}^{2}$

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