for the right triangles below, find the exact values of the side lengths a and c. if necessary, write your…

for the right triangles below, find the exact values of the side lengths a and c. if necessary, write your responses in simplified radical form.

for the right triangles below, find the exact values of the side lengths a and c. if necessary, write your responses in simplified radical form.

Answer

Explanation:

Step1: Solve for side (a) in the first triangle

Use tangent function. (\tan30^{\circ}=\frac{a}{5}). Since (\tan30^{\circ}=\frac{\sqrt{3}}{3}), we have (\frac{\sqrt{3}}{3}=\frac{a}{5}), so (a = \frac{5\sqrt{3}}{3}).

Step2: Solve for side (c) in the second triangle

Use sine - cosine relationships in a (45 - 45-90) triangle. In a (45 - 45 - 90) triangle, the legs are equal and the hypotenuse (c) and a leg (x) are related by (c=\sqrt{2}x). Here (x = 8), so (c=8\sqrt{2}).

Answer:

(a=\frac{5\sqrt{3}}{3}), (c = 8\sqrt{2})