In the figure given below ∠BAC=90⁰ and  AD ⊥ BC, then which of the following options is correct ?       

 

In $\Delta ABD$ and ΔCAD, 

$$\angle ADB=\angle ADC=90⁰$$
$\angle DAB=\angle DAC$, which is common angle
By A.A. similarity criterion
$\Delta ABD ~ \Delta CAD$
Then corresponding sides are in proportion

$\frac{BD}{AD}=\frac{AD}{CD}$

$$AD\mathrm{²} = BD.CD$$
Hence option  $\left(c\right)AD² = BD.CD$is correct