Chapter SH, Problem 12.15EP

To determine

To find: The smallest angles measures for the given triangle.

Answer to Problem 12.15EP

The smallest angle measures is $37°$ .

Explanation of Solution

Given:

The dimensions of the triangle are $4.5$ feet, 6 feet and $7.5$ feet.

Calculation:

Let the triangle is $\Delta ABC$ and $a=7.5$ , $b=6$ and $c=4.5$ .

Use the Law of Cosines $a^2=b^2+c^2-2bc\cos A$ to find the missing side length.

  $\begin{array}{c}{\left(7.5\right)}^2={\left(6\right)}^2+{\left(4.5\right)}^2-2\left(6\right)\left(4.5\right)\cos A\\ 2\left(6\right)\left(4.5\right)\cos A={\left(6\right)}^2+{\left(4.5\right)}^2-{\left(7.5\right)}^2\\ \cos A=0\\ A\approx 90°\end{array}$

Use the Law of Sines to find the missing angles.

  $\begin{array}{c}\frac{\sin B}{6}\approx \frac{\sin 90°}{7.5}\\ \sin B\approx \frac{6\sin 90°}{7.5}\\ B\approx 53°\end{array}$

Find the other angle $C$ .

  $\begin{array}{c}m\angle C\approx 180°-\left(90°+53°\right)\\ \approx 37°\end{array}$

Therefore, the smallest angle measures is $37°$ .