Question

for the right triangles below, find the exact values of the side lengths a and d. the figures are not drawn to scale. (a) a = $\frac{sqrt{2}}{2}$ (b) d = $\frac{1}{2}$

Explanation:

Step1: Solve for side \(a\) in 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\). If the hypotenuse \(c = 1\), and the legs are of length \(a\), using the Pythagorean theorem \(a^{2}+a^{2}=c^{2}\). Since \(c = 1\), we have \(2a^{2}=1\), then \(a^{2}=\frac{1}{2}\), and \(a=\frac{\sqrt{2}}{2}\).

Step2: Solve for side \(d\) in 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the ratio of the sides is \(1:\sqrt{3}:2\). The side opposite the 30 - degree angle is half of the hypotenuse. Here, if the side opposite the 30 - degree angle is the side of length 1, and the side opposite the 90 - degree angle is \(d\), then using the ratio, \(d = \frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).

Answer:

\(a=\frac{\sqrt{2}}{2}\), \(d = \frac{2\sqrt{3}}{3}\)