Question

The area of a triangle is 2312 . Two of the side lengths are 93 and 96 and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of a degree.

Answer 1

to facilitate the exercise we will draw the triangle

We start using the area

$A=\frac{b\times h}{2}$

where A is the area, b the base and h the height

if we replace A=2312 and b=96 we can calculate the height(h)

$\begin{gathered} 2312=\frac{96\times h}{2} \\ \\ h=\frac{2312\times2}{96} \\ \\ h=\frac{289}{6} \end{gathered}$

now to calculate the measure of the angles we can solve the red triangle

first we find Y using trigonometric ratio of the sine

$\sin (\alpha)=\frac{O}{H}$

where alpha is the reference angle, O the opposite side from the angle and H the hypotenuse of the triangle

using Y like reference angle and replacing

$\sin (y)=\frac{\frac{289}{6}}{93}$

simplify

$\sin (y)=\frac{289}{558}$

and solve for y

$\begin{gathered} y=\sin ^{-1}(\frac{289}{558}) \\ \\ y=31.2 \end{gathered}$

value of angle y is 31.2°

Y and X are complementary because make a right line then if we add both numbers the solution is 180°

$\begin{gathered} y+x=180 \\ 31.2+x=180 \end{gathered}$

and solve for x

$\begin{gathered} x=180-31.2 \\ x=148.8 \end{gathered}$

measure of the included angle is 148.8°