Triangle Proportionality theorem
The Triangle Proportionality Theorem
The Triangle Proportionality Theorem is a topic in Similarity in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.8(A), which requires students to prove and apply the proportionality of segments cut by a line parallel to a side of a triangle.
The triangle proportionality theorem states a line parallel to one side of a triangle divides the other two sides proportionally.
Theory
The Triangle Proportionality Theorem (also called the side-splitter theorem) says: a line parallel to one side of a triangle divides the other two sides into proportional segments.
If \(DE\parallel AB\) in \(\triangle ABC\), then \(\dfrac{CD}{DA}=\dfrac{CE}{EB}\).
The converse is also true: if a line divides two sides proportionally, it is parallel to the third side.
The theorem and its converse:
How to use the theorem
- Confirm the line is parallel to a side.
- Match the two pieces of each split side.
- Write the proportion \(\dfrac{CD}{DA}=\dfrac{CE}{EB}\).
- Cross-multiply and solve.
The parallel line divides the sides proportionally.
| \(\dfrac{CD}{DA}\) | \(=\) | \(\dfrac{CE}{EB}\) |
| \(\dfrac{4}{6}\) | \(=\) | \(\dfrac{6}{EB}\) |
| \(4\cdot EB\) | \(=\) | \(36\) |
| \(EB\) | \(=\) | \(9\) |
Cross-multiply and solve.
| \(12x\) | \(=\) | \(48\) |
| \(x\) | \(=\) | \(4\) |
By the converse, equal ratios mean the segment is parallel to the third side.
Equal parts on both sides — the midsegment, which is half the base.
Common pitfalls
Frequently asked questions
What is the triangle proportionality theorem?
A line parallel to one side of a triangle divides the other two sides into proportional segments.
What is the converse?
If a line divides two sides of a triangle proportionally, it is parallel to the third side.
How do you solve for a missing segment?
Set the two side ratios equal in a proportion and cross-multiply.
How does the midsegment relate to this theorem?
The midsegment is the special case where the parallel line passes through the midpoints, splitting both sides equally.