Question

find z.
right triangle with angles 30°, 60°, 90°, hypotenuse 12 mm, side z opposite 60°
write your answer in simplest radical form.
blank millimeters

Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. The hypotenuse is 12 mm. In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse, the side opposite 60° is \(\frac{\sqrt{3}}{2}\) times the hypotenuse, and the side opposite 90° (hypotenuse) is twice the shortest side. Here, \(z\) is opposite the 60° angle.

Step2: Apply 30-60-90 ratios

The hypotenuse \(c = 12\) mm. The side opposite 60° (let's call it \(z\)) is given by \(z=\frac{\sqrt{3}}{2}\times c\). Substitute \(c = 12\):
\(z=\frac{\sqrt{3}}{2}\times12\)
Simplify: \(z = 6\sqrt{3}\)

Answer:

\(6\sqrt{3}\)