Question

D is a point on the side BC of $\Delta ABC$ such that $\angle ADC=\angle BAC$. Then,

• A. $C{B}^2=CA\times CD$
• B. $C{A}^2=CB\times CD$
• C. $C{D}^2=CB\times CA$
• D. None of the above

Answer 1

The correct option is B

$C{A}^2=CB\times CD$

In $\Delta ABC$ and $\Delta DAC$, we have

$\angle ADC=\angle BAC$ [Given]

and $\angle C=\angle C$ [Common angle]

$\therefore \Delta ABC\sim \Delta DAC$ [AA-criterion of similarity]

$\Rightarrow \frac{AB}{DA}=\frac{BC}{AC}=\frac{AC}{DC}$

$\Rightarrow \frac{CB}{CA}=\frac{CA}{DC}$

$\Rightarrow C{A}^2=CB\times CD$.