Side-Splitter Theorem

The side-splitter theorem states that if a line is parallel to one side of a triangle and cross the other two sides, then the line divides those sides proportionally. Take a look at the figure below.

Side-splitter theorem

Since ST is parallel to CB, Then CS / SA = BT / TA

Note that ST is not a midline of the triangle. In other words, S is not the midpoint of CA and T is not the midpoint of BA.

Using the side-splitter theorem to solve a geometric problem.

In the figure below, AB is parallel to UT. Find x if UA = 36, TB = 9, and BS = 3.

Side-splitter theorem exercise

Since AB is parallel to UT, we can use the side-splitter theorem.

UA / AS = TB / BS

Substitute

36 / x = 9 / 3

Solve for x using cross-multiplication 

36 × 3 = x × 9

108 = 9x

Divide both sides by 9

108 / 9 = 9x / 9

12 = x

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