Midsegment Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Midsegment Theorem.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.

Picture a triangular picture frame hanging on a wall. Stretch a rubber band between the midpoints of two sides. That rubber band runs perfectly parallel to the bottom of the frame, like a miniature shelf—and it spans exactly half the width. No matter how you reshape the triangle, that halfway connection always mirrors the opposite side at half scale.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Midpoints create a natural half-scale copy—the midsegment is parallel to and half the length of the opposite side.

Common stuck point: There are three midsegments in every triangle (one for each pair of sides), and together they form the medial triangle, which is similar to the original with scale factor \frac{1}{2}.

Worked Examples

Example 1

easy
In \triangle ABC, M is the midpoint of AB and N is the midpoint of AC. If BC = 18, find MN.

Solution

  1. 1
    Step 1: Identify that MN is the midsegment of \triangle ABC connecting the midpoints of two sides.
  2. 2
    Step 2: By the Midsegment Theorem, the midsegment is parallel to the third side and equal to half its length.
  3. 3
    Step 3: MN = \frac{1}{2} \times BC = \frac{1}{2} \times 18 = 9.

Answer

MN = 9.
The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half as long. This theorem is a special case of similar triangles — the smaller triangle formed is similar to the original with ratio 1:2.

Example 2

medium
In \triangle PQR, M is the midpoint of PQ and N is the midpoint of QR. If MN = 3x - 1 and PR = 4x + 6, find the value of x and the length MN.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The midsegment of a triangle has length 14. What is the length of the side parallel to the midsegment?

Example 2

hard
In \triangle ABC, the three midsegments are drawn, dividing the triangle into four smaller triangles. If the area of \triangle ABC is 120 cm², what is the area of each smaller triangle? Justify using the Midsegment Theorem.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trianglesparallelismsimilarity