Trigonometry Workbook For Dummies
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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.

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Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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