Question

using algebra find the value of each variable. write answers in simplest radical form. 18. diagram with 60° angle, sides labeled 6, p, and m, right angles

Explanation:

Step1: Analyze the congruent right triangles

The figure has two congruent right triangles (shared hypotenuse \(m\), legs 6 and \(p\), one triangle has a \(60^\circ\) angle). In the right triangle with the \(60^\circ\) angle, the side adjacent to \(60^\circ\) is 6, and \(p\) is opposite to \(60^\circ\).
Use \(\tan(60^\circ)=\frac{\text{opposite}}{\text{adjacent}}\)
\(\tan(60^\circ)=\frac{p}{6}\)
Since \(\tan(60^\circ)=\sqrt{3}\), we get \(p = 6\sqrt{3}\)

Step2: Calculate hypotenuse \(m\)

Use the Pythagorean theorem for one right triangle: \(m^2 = 6^2 + p^2\)
Substitute \(p = 6\sqrt{3}\):
\(m^2 = 6^2 + (6\sqrt{3})^2 = 36 + 108 = 144\)
Take the square root: \(m = \sqrt{144} = 12\)

Answer:

\(p = 6\sqrt{3}\), \(m = 12\)