Geometry

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# Similar Triangles

## Definition of Similar Triangles

**Two triangles are considered similar if they have the same shape.** To have the same shape, they must have the same angles and their sides must be in proportion. They do not have to be the same size.

In other words, similar triangles are scaled copies of each other. One could be scaled up by a factor of 2 from the other, scaled down by a factor of 0.35, etc.

**Similar is not the same as congruent.** To be congruent, triangles must have the same shape AND the same size.

### Example Problem

Triangle *ABC* and *DEF* are similar.

Side *AB* = 10

Side *BC* = 8

Side *CA* = 6

Side *DE* = 5

What are the lengths of sides *EF* and *FD*?

Solution:

1.) The triangles are similar but not the same size. Let’s determine the scaling factor between the two, then we can multiply that by the side lengths of triangle *ABC*.

2.) The ratio between the bottom sides is:^{DE}⁄_{AB} = ^{5}⁄_{10} = ^{1}⁄_{2} = 0.5

3.) Therefore, triangle *DEF* is scaled down from triangle *ABC* by a factor of 0.5.

4.) To find side *EF*, we will scale side *BC*.

EF = 0.5BC = 0.5(8) = 4

5.) To find side *FD*, we will scale side *CA*.

FD = 0.5CA = 0.5(6) = 3

6.) **The length of side EF is 4 and the length of side FD is 3.**