You can solve certain similar triangle problems using the Side-Splitter Theorem. This theorem states that if a line is parallel to a side of a triangle and it intersects the other two sides, it divides those sides proportionally. See the below figure.

Check out the following problem, which shows this theorem in action:

Here’s the proof:

Then, because both triangles contain angle *S*, the triangles are similar by AA (Angle-Angle).

Now find *x* and *y*.

And here’s the solution for *y*: First, don’t fall for the trap and conclude that *y* = 4. Side *y* looks like it should equal 4 for two reasons: First, you could jump to the erroneous conclusion that triangle *TRS* is a 3-4-5 right triangle. But nothing tells you that triangle *TRS* is a right angle, so you can’t conclude that.

Second, when you see the ratios of 9 : 3 (along segment *QS*) and 15 : 5 (along segment *PS*, after solving for *x*), both of which reduce to 3 : 1, it looks like *PQ* and *y* should be in the same 3 : 1 ratio. That would make *PQ* : *y* a 12 : 4 ratio, which again leads to the wrong answer that *y* is 4. The answer comes out wrong because this thought process amounts to using the Side-Splitter Theorem for the sides that aren’t split — which you aren’t allowed to do.

Don’t use the Side-Splitter Theorem on sides that aren’t split. You can use the Side-Splitter Theorem *only* for the four segments on the split sides of the triangle. Do not use it for the parallel sides, which are in a different ratio. For the parallel sides, use similar-triangle proportions. (Whenever a triangle is divided by a line parallel to one of its sides, the triangle created is similar to the original, large triangle.)

So finally, the correct way to get *y* is to use an ordinary similar-triangle proportion. The triangles in this problem are positioned the same way, so you can write the following:

That’s a wrap.