Question

Solve triangle ABC using the given measurements if $A=45{}^{\circ },C=90{}^{\circ }\text{and}c=7\sqrt{2}$. Round measure of sides to the nearest tenth and measures of angles to nearest degree.

Short answer

The solution for triangle ABC is$a=7,b=7\text{and}B=45°$.

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 $A=45{}^{\circ },C=90{}^{\circ }\text{and}c=7\sqrt{2}$.

Step 2. Calculation.

In triangle ABC, $A=45{}^{\circ },C=90{}^{\circ }\text{and}c=7\sqrt{2}$. Therefore, $\angle B=45°$ by angle sum property.

Now, trigonometric ratios,

$\begin{array}{c}\sin B=\frac{b}{c}\\ \sin 45°=\frac{b}{7\sqrt{2}}\\ b=7\sqrt{2}\sin \left(45°\right)\\ b=7\end{array}$

Now, in right triangle ABC by Pythagoras Theorem.

$\begin{array}{c}{c}^2=a^2+b^2\text{\hspace{0.17em}gives,}\\ {\left(7\sqrt{2}\right)}^2=a^2+{\left(7\right)}^2\\ a^2={\left(7\sqrt{2}\right)}^2-{\left(7\right)}^2\\ a=\sqrt{{\left(7\sqrt{2}\right)}^2-{\left(7\right)}^2}\\ a=7\end{array}$

Step 3. Conclusion.

Thesolution for triangle ABC, is $a=7,b=7\text{and}B=45°$.