How do you use similar triangles to set up proportions?

You can set up proportions with similar triangles by taking advantage of their side ratios.

By definition, similar triangles have the same angle measures for their corresponding angles, and therefore the corresponding sides have a ratio to them.

For examplle consider the triangles below:

It is given that their corresponding angles have the same measurement, so therefore we can say that they are similar.

Now if we were asked to solve for side ED, then we could do so by setting up a proportion using the side ratios as follows:

##(AC)/(DF) = (CB)/(FE) = (BA)/(DE)##

Now we can just plug in the lengths for the respective sides:

##(7)/(14) = (6)/(12) = (10)/(DE)##

Now we can just simplify and solve:

##1/2 = 1/2 = 10/(DE)##

##10/(DE) = 1/2##

##DE = 2(10) = 20##

This is how we can use proportions to solve for side lengths in similar triangles.

Just make sure you set up your proportions with the corresponding sides, or your ratios might come out wrong.

Just as a fun fact, the fact that side lengths have ratios when the angles are the same is fundamental in trigonometry as well, as you use side ratios there too.

Hope that helped :)