Question

Solve triangle ABC using the given measurements if $a=4,C=90{}^{\circ }\text{and}c=9$. Round measure of sides to the nearest tenth and measures of angles to nearest degree.

Short answer

The solution for triangleABC is$b=8.1,A=26{}^{\circ }\text{and}B=64°$.

Step-by-step solution

Step 1. Write down the given parameters of triangle.

The given triangle ABC is shown below whose parameters are represented as $a=4,C=90{}^{\circ }\text{and}c=9$.

Step 2. Calculation.

In triangle ABC, $a=4,C=90{}^{\circ }\text{and}c=9$.

Now, in right triangle ABC,

$\begin{array}{c}{c}^2=a^2+b^2\text{\hspace{0.17em}gives,}\\ 9^2=4^2+b^2\\ b^2=9^2-4^2\\ b=\sqrt{9^2-4^2}\\ b=8.1\end{array}$

Now by trigonometric ratios,

$\begin{array}{c}\cos B=\frac{a}{c}\\ \cos B=\frac{4}{9}\\ B={\cos }^{-1}\left(\frac{4}{9}\right)\\ B=64°\end{array}$

Now, $C=90{}^{\circ }\text{and}B=64°$, therefore by angle sum property $A=26°$.

Step 3. Conclusion.

Thesolution for triangle ABC, is $b=8.1,A=26{}^{\circ }\text{and}B=64°$.