Midsegment theorem
The Triangle Midsegment Theorem
The Triangle Midsegment Theorem is a topic in Triangle Theorems in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.6(D), which requires students to verify theorems about the midsegments of triangles and apply them.
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.