Question

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

Explanation:

Step1: Find the value of \(c\)

In a \(30 - 60-90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). The side opposite the \(60^{\circ}\) angle is \(6\). Let the side opposite \(30^{\circ}\) (which is \(c\)) be \(x\). Then \(\tan60^{\circ}=\frac{6}{c}\), and also using the ratio, if the side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the side opposite \(30^{\circ}\). So \(c=\frac{6}{\sqrt{3}} = 2\sqrt{3}\).

Step2: Find the value of \(b\)

In a \(45 - 45-90\) triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Here the legs are equal. Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), and since \(a = 3\) (one leg) and \(b\) is the hypotenuse. \(b=\sqrt{3^{2}+3^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).

Answer:

\(c = 2\sqrt{3}\)
\(b = 3\sqrt{2}\)