Choose the one alternative that best completes the statement or answers the question.
Solve the triangle.
-$$\mathrm { a } = 4 , \mathrm { c } = 3 , \beta = 90 ^ { \circ }$$
A) $b = 5 , \alpha = 36.9 ^ { \circ } , \gamma = 53.1 ^ { \circ }$
B) $b = 6 , \alpha = 53.1 ^ { \circ } , \gamma = 36.9 ^ { \circ }$
C) $b = 4 , \alpha = 36.9 ^ { \circ } , \gamma = 53.1 ^ { \circ }$
D) $b = 5 , \alpha = 53.1 ^ { \circ } , \gamma = 36.9 ^ { \circ }$

Question

Quizplus · Accepted Answer

Given that $\beta = 90^\circ$, this is a right-angled triangle with angle $\beta$ as the right angle. Using Pythagoras' theorem, $a^2 + c^2 = b^2$, where $a = 4$ and $c = 3$, we find $b = 5$. To find the angles, we can use trigonometric ratios. Since $\sin(\alpha) = \frac{opposite}{hypotenuse} = \frac{c}{b} = \frac{3}{5}$, $\alpha$ is approximately $36.9^\circ$. Consequently, $\gamma = 90^\circ - \alpha = 53.1^\circ$.

Choose the One Alternative That Best Completes the Statement or Answers