Application of Similar Triangles

Read this section and watch the videos to see the examples of applications of similar triangles in geometric and real-life problems.

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.