Proportional Segments

The side-splitter and angle-bisector theorems.

The idea

Two theorems produce proportional segments inside a triangle, and each lets three known lengths determine a fourth.

Theorem (Side-Splitter Theorem).

A line parallel to one side of a triangle divides the other two sides proportionally: if it cuts one of them into pieces $a$ and $b$, and the other into the matching pieces $c$ and $d$, then $\dfrac{a}{b} = \dfrac{c}{d}$.

Theorem (Angle-Bisector Theorem).

The bisector of an angle of a triangle splits the opposite side into two pieces whose lengths stand in the ratio of the two sides meeting at that angle.

The side-splitter follows from similar triangles: the parallel line cuts off a smaller triangle whose angles match the whole triangle's, so its sides are the whole's scaled by one common factor, and the pieces keep that ratio. Neither theorem requires knowing the size of any angle.

Ways to work on it

• Walkthrough. How parallel lines and angle bisectors divide a triangle's sides proportionally.
• Proof. See why the angle bisector splits the side in the ratio of the adjacent sides.
• Practice. Solve a side-splitter proportion.
• Hardest. Find the segments an angle bisector cuts from the opposite side.

Not sure where to start? Take the ten-question placement test.