# Question #7dfe9

Oct 7, 2016

$y = 12$

#### Explanation:

As $\overline{B D}$ is the angle bisector of $\angle A B C$, we have $\angle A B D = \angle C B D$. As $\angle B A D = \angle B C D = {90}^{\circ}$ and the interior angles of any triangle sum to ${180}^{\circ}$, we must have $\angle B D C = \angle B D A$.

With that, we can use the angle-side-angle property with the shared side $\overline{B D}$ to say that $\triangle A B D \cong \triangle C B D$. Equating corresponding sides, we get

$A D = C D$

$\implies 3 y + 6 = 5 y - 18$

$\implies 2 y = 24$

$\therefore y = 12$

Oct 7, 2016

$y = 12$.

#### Explanation:

Since, $\vec{B D}$ bisects $\angle A B C$,

$m \angle A B D = \frac{1}{2} m \angle A B C = {20}^{\circ} .$

$\Rightarrow 3 x - 1 = {20}^{\circ} \Rightarrow x = {7}^{\circ} .$

Again, pt. $D$ lies on the $\angle$-bisector of $\angle A B C$,

The $\bot - \mathrm{di} s t . D A = \bot - \mathrm{di} s t . D B$

$\therefore 3 y + 6 = 5 y - 18.$

$\therefore 24 = 2 y \text{, giving, } y = 12$