Question

In the triangle ABC, BD bisects angle B and is perpendicular to AC. If the lengths of the sides of the triangle are expressed in terms of x and y as shown, find the values of x and y.

Answer 1

In $△ABD$ and $△CBD$

$BD=BD$ (common)

$\angle ADB=\angle CDB$ (each ${90}^{\circ }$)

$\angle ABD=\angle CDB$ ( $BD$ bisect $\angle B$)

$△ABD\cong △CDB$ (by ASA)

$\Rightarrow 3x+1=5y-2$ (CPCT)

$\Rightarrow x=\frac{5y-3}{3}$ .....$\left(1\right)$

$\Rightarrow x+1=y+2$ (CPCT)

$\Rightarrow x=y+1$ .........$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$

$\Rightarrow 5y-3=3(y+1)$

$\Rightarrow 5y-3y=3+3$

$\Rightarrow 2y=6$

$\Rightarrow y=3$

put $y=3$ in $\left(2\right)$

$x=3+1$

$\therefore x=4$