# What is the value of x?

Feb 5, 2017

$x = 3$

#### Explanation:

Since $C D$ bisects $\angle A C B$, by Angle bisector theorem, we know that:
$\frac{B D}{A D} = \frac{B C}{A C}$
$\implies \frac{x}{2.25} = \frac{4}{3} , \implies x = \frac{2.25 \cdot 4}{3} = 3$

Alternative solution : using Sine Law
In $\Delta A C D , \frac{2.25}{\sin} a = \frac{3}{\sin} b$ ------------EQ(1)
In $\Delta B C D$, $\frac{x}{\sin} a = \frac{4}{\sin} \left(180 - b\right)$
As $\sin b = \sin \left(180 - b\right)$
$\implies \frac{x}{\sin} a = \frac{4}{\sin} b$ ------------------------EQ(2)

By dividing EQ(2) by EQ(1), we get:
$\frac{x}{2.25} = \frac{4}{3}$
$\implies x = \frac{2.25 \cdot 4}{3} = 3$