30-60-90 Triangle Side Ratios

1. Find the ratios between the three sides of any \(30-60-90\) triangle.


[Figure 4]

From the reference image: \(\triangle D E F\) has sides

\(D E=3, E F=3 \sqrt{3}\), and \(F D=6\). This ratio of \(3: 3 \sqrt{3}: 6\) reduces to \(1: \sqrt{3}: 2\).

The three sides of any 30-60-90 triangle will be in this ratio: \(1: \sqrt{3}: 2\).

2. Find the missing sides of the triangle below.


[Figure 5]

The side opposite the \(30^{\circ}\) angle is the smallest side because \(30^{\circ}\) is the smallest angle. Therefore, the length of \(10\) corresponds to the length of \(1\) in the ratio \(1: \sqrt{3}: 2\). The scale factor is \(10\). The other sides of the triangle will be \(10 \sqrt{3}\) and \(20\), because \(10: 10 \sqrt{3}: 20\) is equivalent to \(1: \sqrt{3}: 2\). \(B C=10 \sqrt{3}\) and \(A C=20\).