Midsegment Theorem Examples: 25 Problems with Answers

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

Common stuck point: The procedure for midsegment theorem is the easy part; the trap is doubling instead of halving. Asking "Does the segment join the exact midpoints of two triangle sides?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the segment join the exact midpoints of two triangle sides?

Worked Examples

Example 1

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

Answer

MN=9.

First step

1
Step 1: Identify that MN is the midsegment of △ABC connecting the midpoints of two sides.

Full solution

  1. 2
    Step 2: By the Midsegment Theorem, the midsegment is parallel to the third side and equal to half its length.
  2. 3
    Step 3: MN=12×BC=12×18=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 △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.

Example 3

easy
A triangle has third side of length 40. The midsegment parallel to that side has length ℓ. Find ℓ and explain why.

Example 4

medium
In △ABC, midsegment DE connects midpoints of AB and AC, parallel to BC. If ∠ADE=70°, find ∠ABC.

Example 5

medium
D is the midpoint of AB, E is the midpoint of AC, DE=8 and is parallel to BC. A line parallel to BC is drawn 3/4 of the way from A to BC. Find its length.

Example 6

hard
A quadrilateral has consecutive midpoints of sides connected to form an inner quadrilateral. By Varignon's theorem, what shape is the inner quadrilateral, and what is its perimeter relative to the original diagonals?

Example 7

hard
In △ABC with BC=24, the midsegment MN parallel to BC is extended (along the line through M and N) to a longer chord through the triangle. Why is the extended chord still less than 24 if it stays parallel to BC and inside the triangle?

Example 8

hard
Using coordinates, prove the Midsegment Theorem for △ABC with A(0,0), B(2b,0), C(2c,2d).

Example 9

challenge
Prove that in any quadrilateral, the four midpoints of the sides form a parallelogram whose area is half the area of the original quadrilateral.

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

Example 3

easy
A triangle's third side has length 24. How long is the midsegment parallel to it?

Example 4

easy
In △ABC, M is the midpoint of AB and N is the midpoint of AC. BC=30. Find MN.

Example 5

easy
A midsegment in a triangle is parallel to which side?

Example 6

easy
The three midsegments of a triangle divide it into how many smaller triangles?

Example 7

medium
In △ABC, D and E are midpoints of AB and AC. DE=3x+2 and BC=7x−6. Find x.

Example 8

medium
The medial triangle of △PQR has area 9 cm2. Find the area of △PQR.

Example 9

medium
In △ABC, the three midsegments divide it into four congruent smaller triangles. If △ABC has area 48 cm2, find the area of each smaller triangle.

Example 10

medium
In △ABC, the midsegment parallel to BC has length 13. Find BC.

Example 11

medium
A triangle has perimeter 36. Find the perimeter of its medial triangle.

Example 12

hard
In △ABC, M is the midpoint of AB and N is on AC with MN∥BC. Prove N is the midpoint of AC.

Example 13

hard
In △ABC, the medial triangle has area 5 cm2. Find the area of each of the three corner triangles formed by the midsegments.

Example 14

hard
A triangle has vertices A(0,0), B(6,0), C(2,4). Find the length of the midsegment parallel to BC.

Example 15

hard
In △ABC, let G be the centroid. A midsegment MN is drawn parallel to BC. What fraction of the way from A to BC does G lie?

Example 16

challenge
The medial triangle of a triangle is itself replaced by its medial triangle (one level deeper). What fraction of the original area is this second-level medial triangle?
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Background Knowledge

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

trianglesparallelismsimilarity