Similar Triangles: Perimeters and Areas

When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. In Figure 1 , Δ ABC∼ Δ DEF.

Figure 1 Similar triangles whose scale factor is 2 : 1.

The ratios of corresponding sides are 6/3, 8/4, 10/5. These all reduce to 2/1. It is then said that the scale factor of these two similar triangles is 2 : 1.

The perimeter of Δ ABC is 24 inches, and the perimeter of Δ DEF is 12 inches. When you compare the ratios of the perimeters of these similar triangles, you also get 2 : 1. This leads to the following theorem.

Theorem 60: If two similar triangles have a scale factor of a : b, then the ratio of their perimeters is a : b.

Example 1: In Figure 2 , Δ ABC∼ Δ DEF. Find the perimeter of Δ DEF

Figure 2 Perimeter of similar triangles.

Figure 3 shows two similar right triangles whose scale factor is 2 : 3. Because GH ⊥ GI and JK ⊥ JL , they can be considered base and height for each triangle. You can now find the area of each triangle.

Figure 3 Finding the areas of similar right triangles whose scale factor is 2 : 3.

Now you can compare the ratio of the areas of these similar triangles.

This leads to the following theorem:

Theorem 61: If two similar triangles have a scale factor of a : b, then the ratio of their areas is a² : b².

Example 2: In Figure 4 , Δ PQR∼ Δ STU. Find the area of Δ STU.

Figure 4 Using the scale factor to determine the relationship between the areas of similar triangles.

The scale factor of these similar triangles is 5 : 8.

Example 3: The perimeters of two similar triangles is in the ratio 3 : 4. The sum of their areas is 75 cm². Find the area of each triangle.

If you call the triangles Δ₁ and Δ₂, then

According to Theorem 60, this also means that the scale factor of these two similar triangles is 3 : 4.

Because the sum of the areas is 75 cm², you get

Example 4: The areas of two similar triangles are 45 cm² and 80 cm². The sum of their perimeters is 35 cm. Find the perimeter of each triangle.

Call the two triangles Δ₁ and Δ₂ and let the scale factor of the two similar triangles be a : b.

a : b is the reduced form of the scale factor. 3 : 4 is then the reduced form of the comparison of the perimeters.

Reduce the fraction.

Take square roots of both sides.

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