Question

1. find the area and perimeter of the following triangle. show all work.

Explanation:

Step1: Find the other leg

Use trigonometric ratio $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the other leg be $x$. Given $\theta = 47^{\circ}$ and adjacent side $a = 19$.
$\tan47^{\circ}=\frac{x}{19}$, so $x = 19\times\tan47^{\circ}\approx19\times1.0724 = 20.3756$.
Use $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Let hypotenuse be $h$. $\cos47^{\circ}=\frac{19}{h}$, so $h=\frac{19}{\cos47^{\circ}}\approx\frac{19}{0.6820}\approx27.86$.

Step2: Calculate the area

Area of a right - triangle $A=\frac{1}{2}\times\text{base}\times\text{height}$. Here, base $b = 19$ and height $h_{t}=20.3756$.
$A=\frac{1}{2}\times19\times20.3756= 193.5682$.

Step3: Calculate the perimeter

Perimeter $P=\text{sum of all sides}$. $P=19 + 20.3756+27.86=67.2356$.

Answer:

The area is approximately $193.57$ and the perimeter is approximately $67.24$.