Proportionality Theorems

When studying similar triangles, certain geometric setups create predictable proportional relationships between line segments. Here are the three most important proportionality theorems you need to know.

Triangle Proportionality Theorem (Side-Splitter Theorem)

If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.

If line segment $DE$ is parallel to side $BC$ ($DE \parallel BC$), then: $\frac{AD}{DB} = \frac{AE}{EC}$

Example: In $\triangle ABC$, $DE \parallel BC$ with $AD = 4$, $DB = 6$, and $AE = 5$. Find $EC$.

Solution: Set up the proportion using the theorem: $\frac{4}{6} = \frac{5}{EC}$ Cross-multiply to solve for $EC$: $4 \cdot EC = 6 \cdot 5$ $4 \cdot EC = 30$ $EC = 7.5$

Triangle Angle Bisector Theorem

An angle bisector of a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle.

If line segment $AD$ bisects $\angle A$ in $\triangle ABC$, intersecting side $BC$ at point $D$, then: $\frac{BD}{DC} = \frac{AB}{AC}$

Geometric Mean (Altitude) Theorem

In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of those two segments.

If $CD$ is the altitude drawn to the hypotenuse $AB$ of right $\triangle ABC$, then: $\frac{AD}{CD} = \frac{CD}{DB} \implies CD^2 = AD \cdot DB$

Example: In right $\triangle ABC$ with altitude $CD$ to the hypotenuse, if $AD = 3$ and $DB = 12$, find $CD$.

Solution: Using the geometric mean theorem: $CD^2 = AD \cdot DB$ $CD^2 = 3 \cdot 12$ $CD^2 = 36$ Take the square root of both sides: $CD = 6$