Similar Triangles

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

These triangles are all similar:

triangles similar different sizes and rotations

(Equal angles have been marked with the same number of arcs)

Some of them have different sizes and some of them have been turned or flipped.

 

For similar triangles:

corresponding angles on two triangles
All corresponding angles are equal

corresponding sides on two triangles
All corresponding sides have the same ratio

Also notice that the corresponding sides face the corresponding angles. For example the sides that face the angles with two arcs are corresponding.

Corresponding Sides

In similar triangles, corresponding sides are always in the same ratio.

For example:

triangles similar R: (6,7,8) and S: (b,a,6.4)

Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

Calculating the Lengths of Corresponding Sides

We can sometimes calculate lengths we don't know yet.

Example: Find lengths a and b of Triangle S

triangles similar R: (6,7,8) and S: (b,a,6.4)

Step 1: Find the ratio

We know all the sides in Triangle R, and
We know the side 6.4 in Triangle S

The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R.

So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is:

6.4 to 8

Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R.

Step 2: Use the ratio

a faces the angle with one arc as does the side of length 7 in triangle R.

a = (6.4/8) × 7 = 5.6

 

b faces the angle with three arcs as does the side of length 6 in triangle R.

b = (6.4/8) × 6 = 4.8

 

Done!

thumb distance far

Did You Know?

Similar triangles can help you estimate distances.