# Solve for x? Show the equation you used and your work.

Jan 18, 2017

$x = 27$.

#### Explanation:

Since the Ray $A D$ bisects the $\angle B A C$, we know from Geometry,

that, $\frac{B D}{D C} = \frac{A B}{A C} \Rightarrow \frac{24}{18} = \frac{36}{x} \Rightarrow x = \frac{\left(18\right) \left(36\right)}{24} = 27$

Jan 18, 2017

See below.

#### Explanation:

Using the so called sinus law

$\Delta A B D \to \sin \frac{\angle \frac{B A C}{2}}{\overline{B D}} = \sin \frac{\angle B D A}{\overline{B A}}$
$\Delta A D C \to \sin \frac{\angle \frac{B A C}{2}}{\overline{D C}} = \sin \frac{\angle A D C}{x}$

but $\angle B D A + \angle A D C = \pi$ so $\sin \left(\angle B D A\right) = \sin \left(\angle A D C\right)$

Dividing the two relationships term to term we get

$\frac{\overline{D C}}{\overline{B D}} = \frac{x}{\overline{B A}}$ so

$x = \frac{\overline{B A} \overline{D C}}{\overline{B D}} = \frac{36 \cdot 18}{24} = 27$