Mathwords logoReference LibraryMathwords

Midsegment

A midsegment is a line segment that connects the midpoints of two sides of a triangle. It is always parallel to the third side and exactly half as long.

In a triangle △ABC\triangle ABC, a midsegment is the segment joining the midpoint of one side to the midpoint of another side. By the Triangle Midsegment Theorem, this segment is parallel to the third side and its length equals one-half the length of that third side. Every triangle has exactly three midsegments, one for each pair of sides.

Key Formula

m=12⋅sm = \frac{1}{2} \cdot s
Where:
  • mm = the length of the midsegment
  • ss = the length of the third side (the side the midsegment is parallel to)

Worked Example

Problem: In triangle ABC, the midpoint of side AB is M and the midpoint of side AC is N. If side BC has a length of 18 cm, find the length of midsegment MN.
Step 1: Identify which side the midsegment is parallel to. Since M is the midpoint of AB and N is the midpoint of AC, the midsegment MN is parallel to side BC.
MN∥BCMN \parallel BC
Step 2: Apply the Triangle Midsegment Theorem. The midsegment is half the length of the side it is parallel to.
MN=12⋅BCMN = \frac{1}{2} \cdot BC
Step 3: Substitute the known length of BC and calculate.
MN=12⋅18=9 cmMN = \frac{1}{2} \cdot 18 = 9 \text{ cm}
Answer: The midsegment MN has a length of 9 cm.

Why It Matters

Midsegments show up frequently in geometric proofs and coordinate geometry problems. Engineers and architects use them when working with triangular structures, since knowing one measurement lets you quickly determine another. They also provide an accessible introduction to the idea of similar figures, because the midsegment creates a smaller triangle that is similar to the original.

Common Mistakes

Mistake: Confusing a midsegment with a median. A median goes from a vertex to the midpoint of the opposite side, while a midsegment connects the midpoints of two sides.
Correction: Remember that a midsegment has no endpoint at a vertex — both of its endpoints are midpoints of sides.
Mistake: Doubling instead of halving. Some students set the midsegment equal to twice the third side rather than half.
Correction: The midsegment is always the shorter segment. If BC=18BC = 18, then the midsegment parallel to BC is 99, not 3636.

Related Terms