In the given figure, =90^ and AD . Then: - Vidyadip

MCQ

In the given figure, $\angle\text{BAC}=90^\circ$ and $\text{AD}\perp\text{BC}.$ Then:

A
$\text{BC}\cdot\text{CD}=\text{BC}^2$
B
$\text{AB}\cdot\text{AC}=\text{BC}^2$
✓
$\text{BD}\cdot\text{CD}=\text{AD}^2$
D
$\text{AB}\cdot\text{AC}=\text{AD}^2$

✓

Answer

Correct option: C.

$\text{BD}\cdot\text{CD}=\text{AD}^2$

In $\triangle\text{ABC},$
$\angle\text{ABD}=90^\circ-\angle\text{C}$
Similarly, in $\triangle\text{ACD},$
$\angle\text{CAD}=90^\circ-\angle\text{C}$
In $\triangle\text{DBA}$ and $\triangle\text{DAC}$
$\angle\text{ADB}=\angle\text{CDA}=90^\circ$
$\angle\text{ABD}=\angle\text{CAD}=90^\circ-\angle\text{C}$
So, $\triangle\text{DBA}\sim\triangle\text{DAC}$ .....(AA criterion of similarity)
$\frac{\text{BD}}{\text{AD}}=\frac{\text{AD}}{\text{CD}}$
$\Rightarrow\text{BD}\cdot\text{CD}=\text{AD}^2$

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