Application of Similar Triangles
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 \(A C=20\).