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: 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

\triangle ABC

M

is the midpoint of

AB

and

N

is the midpoint of

AC

. If

BC = 18

, find

MN

M is the midpoint of AB and N is the midpoint of AC. Find MN.

Answer

MN = 9

First step

Step 1: Identify that

MN

is the midsegment of

\triangle ABC

connecting the midpoints of two sides.

Full solution

2
Step 2: By the Midsegment Theorem, the midsegment is parallel to the third side and equal to half its length.
3
Step 3: $MN = \frac{1}{2} \times BC = \frac{1}{2} \times 18 = 9$ .

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

\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

M is the midpoint of PQ and N is the midpoint of QR. MN = 3x − 1; PR = 4x + 6.

Example 3

easy

A triangle has third side of length

40

. The midsegment parallel to that side has length

\ell

. Find

\ell

and explain why.

Example 4

medium

\triangle ABC

, midsegment

DE

connects midpoints of

AB

and

AC

, parallel to

BC

. If

\angle ADE = 70°

, find

\angle 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

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

\triangle 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

\triangle 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?

MN is the midsegment parallel to BC. MN = 14; find BC.

Example 2

hard

\triangle ABC

, the three midsegments are drawn, dividing the triangle into four smaller triangles. If the area of

\triangle ABC

120

cm², what is the area of each smaller triangle? Justify using the Midsegment Theorem.

The three midsegments divide triangle ABC (area 120 cm²) into four congruent triangles. Find the area of each.

Example 3

easy

A triangle's third side has length

24

. How long is the midsegment parallel to it?

Example 4

easy

\triangle 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

\triangle 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

\triangle PQR

has area

9

^2

. Find the area of

\triangle PQR

Example 9

medium

\triangle ABC

, the three midsegments divide it into four congruent smaller triangles. If

\triangle ABC

has area

48

^2

, find the area of each smaller triangle.

Example 10

medium

\triangle 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

\triangle ABC

M

is the midpoint of

AB

and

N

is on

AC

with

MN \parallel BC

. Prove

N

is the midpoint of

AC

Example 13

hard

\triangle ABC

, the medial triangle has area

5

^2

. 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

\triangle ABC

, let

G

be the centroid. A midsegment

MN

is drawn parallel to

BC

. What fraction of the way from

A

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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Related Concepts

Triangles Parallelism Similarity Coordinate Proofs Proportional Geometry

Background Knowledge

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

trianglesparallelismsimilarity