find the perimeter of the triangle to the right.\nthe perimeter is \n(type an integer or decimal rounded to…

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is $180^{\circ}$. Let the third angle be $x$. Then $x = 180-(48 + 68)=64^{\circ}$.

Step2: Use the Law of Sines

Let the sides opposite the angles $48^{\circ}$, $68^{\circ}$, and $64^{\circ}$ be $a$, $b = 23$, and $c$ respectively. By the Law of Sines, $\frac{a}{\sin48^{\circ}}=\frac{b}{\sin68^{\circ}}=\frac{c}{\sin64^{\circ}}$. First, find $a$: $a=\frac{b\sin48^{\circ}}{\sin68^{\circ}}=\frac{23\sin48^{\circ}}{\sin68^{\circ}}\approx\frac{23\times0.7431}{0.9272}\approx18.5$. Then find $c$: $c=\frac{b\sin64^{\circ}}{\sin68^{\circ}}=\frac{23\sin64^{\circ}}{\sin68^{\circ}}\approx\frac{23\times0.8988}{0.9272}\approx22.2$.

Step3: Calculate the perimeter

The perimeter $P=a + b + c$. Substitute $a\approx18.5$, $b = 23$, and $c\approx22.2$ into the formula. $P\approx18.5+23 + 22.2=63.7$.

Answer:

$63.7$