Triangles: Learn It 3

Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.

The two triangles below are similar. Each side of [latex]\Delta ABC[/latex] is four times the length of the corresponding side of [latex]\Delta XYZ[/latex] and their corresponding angles have equal measures.

[latex]\Delta ABC[/latex] and [latex]\Delta XYZ[/latex] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.

 

properties of similar triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

 

Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled b, and the side across from C is labeled c. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled y, and the side across from Z is labeled z. The vertices are labeled with capital letters and the sides are labeled with lowercase letters. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is a proportion of only lowercase letters reading a over x equals b over y equals c over z.

 

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in [latex]\Delta ABC\text{:}[/latex]

[latex]\begin{array}{c}\text{the length }a\text{ can also be written }BC\hfill \\ \text{the length }b\text{ can also be written }AC\hfill \\ \text{the length }c\text{ can also be written }AB\hfill \end{array}[/latex]

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

We can use the notion that triangles are similar to solve for an unknown side length.

How To: Find the Unknown Side Length in a Triangle When Given Similar Triangles

To find an unknown side length in a triangle when given similar triangles, you can use the concept of proportional relationships between corresponding sides. Below are steps to follow when solving for an unknown side given similar triangles. We will use the following triangles to demonstrate these steps:

 

two similar triangles. Triangle ABC has sides labeled AB 18, BC 12, and CA x. Triangle EDF es labeled DE y, EF 9, and FD 18.. Angle A is equal to angle D, angle B is equal to angle E and angle C is equal to angle F.

 

  1. Identify the corresponding sides in the two similar triangles.
    • [latex]AB \sim ED, BC \sim EF, CA \sim FD[/latex]
  2. Write the ratio of the lengths of the corresponding sides.
    • [latex]\frac{AB}{ED} = \frac{BC}{EF} = \frac{CA}{FD}[/latex]
  3. Set up a proportion to solve for the unknown.
    • [latex]\frac{AB}{ED} = \frac{BC}{EF}[/latex]
    • [latex]\frac{18}{x} = \frac{12}{9}[/latex]
  4. Solve the proportion by cross multiplying and dividing to find the unknown
    • [latex]AB*EF = BC*ED[/latex]
    • [latex]18*9 = 12*x[/latex]
    • [latex]\frac{18*9}{12} =x[/latex]

Here is an example for you to try.

[latex]\Delta ABC[/latex] and [latex]\Delta XYZ[/latex] are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.