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 *a*^{2} : *b*^{2}.

**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 cm^{2}. Find the area of each triangle.

If you call the triangles Δ_{1} and Δ_{2}, 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 cm^{2}, you get

**Example 4:** The areas of two similar triangles are 45 cm^{2} and 80 cm^{2}. The sum of their perimeters is 35 cm. Find the perimeter of each triangle.

Call the two triangles Δ_{1} and Δ_{2} 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.