The Triangle Midsegment Theorem

A midsegment of a triangle is the segment that connects the midpoints of two sides. Every triangle has three of them, and each one obeys the same two-part rule: a midsegment is parallel to the third side of the triangle, and it is exactly half as long.

That "half as long" fact does all the work in problems. Given the third side, halve it to get the midsegment. Given the midsegment, double it to get the third side. The only real skill is recognizing which segment is the midsegment and which is the third side.

What a midsegment is

In $\triangle ABC$ , let $M$ be the midpoint of $\overline{AB}$ and $N$ be the midpoint of $\overline{AC}$ . Then $\overline{MN}$ is a midsegment. In diagrams, midpoints are usually shown with tick marks — one tick on each half of a bisected side.

The third side is the side the midsegment never touches. For $\overline{MN}$ above, that is $\overline{BC}$ . The theorem connects exactly those two segments: $MN \parallel BC$ and $MN = \dfrac{1}{2} \cdot BC$ .

In the figure, the tick marks show $M$ and $N$ bisect their sides, and the matching chevrons show $\overline{MN} \parallel \overline{BC}$ .

Using the rule in both directions

Midsegment from side: halve. If $BC = 14$ , then $MN = 7$ .

Side from midsegment: double. If $MN = 9$ , then $BC = 18$ .

The most common error is halving when you should double. Before you compute, ask: is the segment I want the short one (the midsegment) or the long one (the third side)? The midsegment is always the smaller of the two.

Why it works

A midsegment is a side-splitter taken to the extreme: it cuts both sides in the ratio $1:1$ . That makes the small triangle above the midsegment similar to the whole triangle with scale factor $\dfrac{1}{2}$ — which is exactly where "parallel to the third side and half its length" comes from.

This connection is worth remembering. If a segment joins two points on the sides of a triangle but they are not both midpoints, the midsegment theorem does not apply — you are back to the general side-splitter setup.

Worked examples

Example 1: find the midsegment

In $\triangle ABC$ , $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ . If $BC = 14$ , find $MN$ .

\overline{MN}

joins two midpoints, so it is a midsegment

Apply the Triangle Midsegment Theorem

MN = \dfrac{1}{2} \cdot BC

Substitute

BC = 14

MN = \dfrac{1}{2} \cdot 14 = 7

Answer: $MN = 7$

Example 2: find the third side

$\overline{MN}$ is a midsegment of $\triangle ABC$ and $MN = 9$ . Find $BC$ .

Write the theorem

MN = \dfrac{1}{2} \cdot BC

Substitute

MN = 9

9 = \dfrac{1}{2} \cdot BC

Multiply both sides by

2

BC = 18

Answer: $BC = 18$

Example 3: a midsegment with algebra

A midsegment has length $3x$ and the parallel third side has length $42$ . Find $x$ .

The midsegment is half the third side

3x = \dfrac{1}{2} \cdot 42

Simplify the right side

3x = 21

Divide both sides by

3

x = 7

Answer: $x = 7$

Try one yourself

Common questions

Which side is the midsegment half of?

The side it does not touch — the third side. A midsegment connects the midpoints of two sides, and it is parallel to and half the length of the remaining side. It is not half of either side it actually touches.

How many midsegments does a triangle have?

Three — one for each pair of sides. Together they form a smaller triangle inside the original, called the midsegment triangle, and each of its sides is half the length of a side of the original.

What if the segment connects a midpoint to a point that is not a midpoint?

Then it is not a midsegment and the halving rule does not apply. If the segment is parallel to the third side, use the Triangle Proportionality (side-splitter) Theorem instead — the midsegment theorem is just its $1:1$ special case.

How do I spot midpoints in a diagram?

Look for tick marks. A single tick on both pieces of a side means those pieces are congruent, so the point between them is a midpoint. Two different sides bisected — with matching ticks on each — is the signature of a midsegment problem.

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