Education.com
Try
Brainzy
Try
Plus

# Trigonometry and Triangles Study Guide (page 2)

(not rated)
By
Updated on Oct 1, 2011

## Similar Triangles

A triangle is scaled, magnified, or enlarged if the length of each side is multiplied by the same number. For example, in Figure 2.6, ΔDEF is a magnified version of ΔABC because the three sides of ΔDEF are twice as big as the three sides of ΔABC. The scale factor in this example is thus 2.

We can construct ΔDEF with four copies of ΔABC as illustrated in Figure 2.7.

This means that the area of ΔDEF is four times the area of ΔABC . It also means that the angles of ΔDEF are the same as the angles of ΔABC.

If a triangle is enlarged by a scale factor of 3, then the new triangle will have sides three times longer, will have nine times the area, and will have all the same angles. The most important detail for us is that the angles remain the same. Two triangles are similar if they have the same three angles.

A scaled version of a triangle is always similar to the original, no matter what the scale factor may be. If each side of the second triangle is k times longer than each side in the first, then the angles are all the same. This is depicted in Figure 2.8.

It can also be proven that if two triangles are similar, then one of the triangles is a scaled version of the other. This is illustrated in Figure 2.9. Because the triangles are similar, the sides of the second triangle must be some multiple k of the sides from the first triangle.

#### Example

Suppose there is a triangle with sides 8, 9, and 11 inches in length. Another triangle has the same angles θ, α, and β, but the side between angles α and θ has a length of 6 inches. What are the lengths x and y of the other two sides of the triangle, as shown in Figure 2.10?

Because the two triangles have all the same angles, they are similar. Thus, the sides of the second triangle are some multiple k of the sides of the first triangle.

6 = k · 9
x = k · 8
y = k · 11

The first equation can be solved k = . Thus,

Notice that if two triangles have two angles in common, then they must have all three angles in common, because the third angle must make up the difference to 180°. This is shown in Figure 2.11.

Practice problems for this study guide can be found at:

Trigonometry and Triangles Practice Questions

150 Characters allowed

### Related Questions

#### Q:

See More Questions

### Today on Education.com

Top Worksheet Slideshows