Midsegment theorem
The Triangle Midsegment Theorem
The Triangle Midsegment Theorem is a topic in Triangle Theorems in the California Common Core State Standards. It is aligned to Standard G-CO.10, which requires students to prove theorems about triangles, including that the segment joining midpoints of two sides is parallel to the third and half its length.
A midsegment of a triangle joins the midpoints of two sides and is parallel to the third side and half its length.
Theory
A midsegment of a triangle joins the midpoints of two sides. The Midsegment Theorem states two things at once:
- the midsegment is parallel to the third side, and
- the midsegment is half the length of that third side.
Joining all three midpoints creates the medial triangle, which splits the original into four congruent triangles.
The midsegment relationships:
How to use the midsegment theorem
- Confirm the segment joins two midpoints.
- Length: midsegment \(=\dfrac12\times\) third side (or third side \(=2\times\) midsegment).
- Direction: the midsegment is parallel to the third side.
A midsegment is half the length of the parallel side.
| \(\dfrac{14}{2}\) | \(=\) | \(7\) |
The side is twice the midsegment.
| \(2\times 6\) | \(=\) | \(12\) |
The side equals twice the midsegment.
| \(3x-1\) | \(=\) | \(2(x+3)\) |
| \(3x-1\) | \(=\) | \(2x+6\) |
| \(x\) | \(=\) | \(7\) |
The midsegment is parallel to the third side.
| \(\text{midsegment}\) | \(\parallel\) | \(\text{third side}\) |
Common pitfalls
Frequently asked questions
What is a midsegment of a triangle?
A segment connecting the midpoints of two sides of the triangle.
What does the midsegment theorem state?
A midsegment is parallel to the third side and exactly half its length.
How do you find the third side from the midsegment?
Double the midsegment: the third side is twice as long.
What is the medial triangle?
The triangle formed by the three midsegments; it divides the original into four congruent triangles.