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

Explanation:

Step1: Find side - length (a) using trigonometry

In a (30 - 60-90) right - triangle, if the hypotenuse (c = 2) and we want to find the side opposite the (30^{\circ}) angle ((a)). The ratio of the sides in a (30 - 60-90) triangle is (1:\sqrt{3}:2) (where the side opposite the (30^{\circ}) angle is (x), the side opposite the (60^{\circ}) angle is (\sqrt{3}x), and the hypotenuse is (2x)). Here, (a=\frac{1}{2}\times) hypotenuse. So (a = 1).

Step2: Find side - length (h) using trigonometry

In a (45 - 45-90) right - triangle, the ratio of the sides is (1:1:\sqrt{2}). If the hypotenuse (c = 3), and we know that for a (45 - 45-90) triangle (c=\sqrt{2}s) (where (s) is the length of each of the two equal legs, and in this case (s = h)). Then (h=\frac{3}{\sqrt{2}}). Rationalizing the denominator, we multiply the numerator and denominator by (\sqrt{2}), so (h=\frac{3\sqrt{2}}{2}).

Answer:

(a = 1) (h=\frac{3\sqrt{2}}{2})