Choose the correct answer from the given four options:
In the figure below, $ \angle \mathrm{BAC}=90^{\circ} $ and $ \mathrm{AD} \perp \mathrm{BC} $. Then,

(A) $ \mathrm{BD} \cdot \mathrm{CD}=\mathrm{BC}^{2} $
(B) $ \mathrm{AB}

Given:

\( \angle \mathrm{BAC}=90^{\circ} \) and \( \mathrm{AD} \perp \mathrm{BC} \).

To do:

We have to choose the correct answer.

Solution:

In $\triangle ADB$ and $\triangle ADC$,

$\angle D=\angle D=90^o$

$\angle DBA=\angle DAC=90^0-\angle C$

Therefore, by AA similarity,

$\triangle ADB \sim \triangle ADC$

This implies,

$\frac{BD}{AD}=\frac{AD}{CD}$          (CPCT)

$BD . CD = AD^2$

Hence, \( \mathrm{BD} \cdot \mathrm{CD}=\mathrm{BC}^{2} \) is the correct option.

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Updated on: 10-Oct-2022

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