Given:
The sides of the right triangle are given as,
[tex]\begin{gathered} a=30.4 \\ c=50.2 \end{gathered}[/tex]The objective is to find the other side and the angles of the right triangle.
Explanation:
To find b:
The third side of the right triangle can be calculated using the Pythagorean theorem.
[tex]\begin{gathered} b^2=c^2-a^2 \\ b=\sqrt[]{c^2-a^2}\text{ . . . . . . (1)} \end{gathered}[/tex]On plugging the values of c and a in equation (1),
[tex]b=\sqrt[]{50.2^2-30.4^2}[/tex]On further solving the above equation,
[tex]\begin{gathered} b=\sqrt[]{2520.04-924.16} \\ =\sqrt[]{1595.88} \\ =39.9484668\ldots\ldots \\ \approx39.95 \end{gathered}[/tex]Thus, the side b of the right triangle is 38.95.
To find angle m∠C:
Since the given figure contains a right triangle, the measure of angle m∠C will be 90°.
To find angle m∠A:
The measure of angle m∠A can be calculated using the trigonometric ratio of sin.
[tex]\begin{gathered} \sin A=\frac{a}{c} \\ A=\sin ^{-1}(\frac{a}{c})\text{ . . . . . .(2)} \end{gathered}[/tex]On plugging the obtained values in equation (2),
[tex]\begin{gathered} A=\sin ^{-1}(\frac{30.4}{50.2}) \\ =37.27042\ldots\text{..} \\ \approx37.3\degree \end{gathered}[/tex]To find angle m∠B:
Using the angle sum property of the right triangle, the measure of angle B can be calculated as,
[tex]\begin{gathered} A+B+C=180\degree \\ B=180\degree-A-C\text{ . . . . (3)} \end{gathered}[/tex]On plugging the obtained values in equation (3),
[tex]\begin{gathered} B=180-37.3-90 \\ B=52.7\degree \end{gathered}[/tex]Hence, the sides and the angles of the right triangle are,
a = 30.4
b = 39.95
c = 50.2
m∠A = 37.3°
m∠B = 52.7°
m∠C = 90°
