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All about the Midsegment of a Triangle

DIANE ROCHA, Elizabeth Often
  • Author
    DIANE ROCHA

    Diane has taught High School Math for over fourteen years. She has a BA in Secondary Education - Mathematics from Rivier University and a MA in Education from Worcester State University. Diane's passions are teaching Algebra 1 and Algebra 2 and is currently teaching Precalculus.

  • Instructor
    Elizabeth Often

    Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

Learn how to find the midsegment of a triangle. See the midsegment definition, how the formula works, and find the length of the midsegment by reviewing an example. Updated: 01/24/2022

Midsegment

What is the Midsegment of a Triangle?

Midsegment Definition: The midsegment of a triangle is the segment that connects the midpoints of two sides of a triangle. The midsegment of the triangle is parallel to the third side of the triangle.

Midsegment of a Triangle

Midsegment of a triangle image

In the first image, segment DE is a midsegment of triangle ABC because it is the segment that connects the midpoints of AB and AC. This midsegment of this triangle is parallel to side BC.

Because all three sides have midpoints, more than one midsegment is in each triangle.

Example of a triangle with all three midsegments drawn

Triangle with all three midsegments shown

That means that each triangle will have three midsegments! One midsegment is parallel to each of the three sides of the triangle.

Midsegment Theorem

The Midsegment Theorem states: If a segment joins the midpoints of two sides of a triangle, the segment is parallel to the third side and is half as long.

The unique characteristics of a midsegment of a triangle are that the midsegment connects the midpoints of two sides of a triangle. Another unique characteristic is that the midsegment of a triangle is always parallel to the third side of the triangle.

Definition

The midsegment of a triangle is defined as the segment formed by connecting the midpoints of any two sides of a triangle. Put simply, it divides two sides of a triangle equally. The midpoint of a side divides the side into two equal segments. As you can see in the picture below, DE is the midsegment of the triangle ABC. Point D divides segment AB into two equal parts, and point E divides segment CB into two equal parts. The side of the triangle that the midsegment does not intersect is the base of the triangle.

In this triangle, midsegment DE divides segment AB into two equal parts, and divides segment CB into two equal parts.
Triangle showing midsegment

The midsegment of a trapezoid is the segment formed by connecting the midpoints of the two legs of a trapezoid. Although the midsegment of a trapezoid is also useful in mathematics, we will not discuss it here.

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Midsegment Formula

The midsegment of a triangle formula is:

Midsegment=1/2 the base of the triangle.

The midsegment formula is derived from the fact that by creating a new triangle within the original triangle by taking the midpoints of the two sides, it is creating a triangle that is ultimately half of the original triangle because the triangles are similar triangles by Angle-Angle similarity since the midsegment and the third side are parallel to each other.

Find the Length of the Midsegment of a triangle

Here is an example of how to find the length of the midsegment of a triangle:

Find the measure of midsegment DE

Example of Midsegment Theorem

In order to solve this problem, it is important to know that side BC measures 10 units and 1/2 (10) = 5, the measure of the Midsegment of the triangle (DE) is 5 units.

Practice Problems

Practice Problem 1

Practice problem 1

Practice problem involving finding the measure of the third side when the midsegment is known

Practice Problem 2

Find the value of x.

Finding the value of a midsegment when a variable is shown

Practice Problem 3

Example involving Algebraic expressions

The third practice problem, involving algebra

Solutions

The solutions to Practice Problems 1-3 and explanations for how to solve to find the length of the midsegment of a triangle are below:

Practice Problem 1:

DE is the midsegment, and BC is the side parallel to the midsegment. DE = 9 and DE = 1/2(BC), therefore 9=1/2(BC) so BC is 18.


Practice Problem 2:

DE is the Midsegment, and BC is the side parallel to the Midsegment. DE=x and BC=16, DE=1/2(BC) Therefore, x=1/2(16) so x=8. The measure of Midsegment DE is 8.


Practice Problem 3: DE is the Midsegment and BC is the side parallel to the Midsegment. DE=3x+1 and BC = 4x+8, DE=1/2(BC)

{eq}3x+1=1/2(4x+8) 3x+1 = 2x+4 x=3 {/eq}

Therefore, when we set up and solve the equation using the midsegment formula, we find x=3.

How to Find the Midsegment of a Triangle


Constructing the Midsegment

Image of a compass


The midsegment of a triangle can be found using a compass. We will need the midpoint of two sides of a triangle in order to find the midsegment with a compass.


Properties of the Midsegment and Midsegment Theorem

The triangle midsegment has several useful properties:

  • The midsegment is half the length of the base

The midsegment is one-half the length of the base.
Midsegment labeled with length

  • The midsegment is parallel to the base
  • The triangle formed by the midsegment and the two half sides have all the same angle measures as the original triangle
  • The perimeter of the triangle formed by the midsegment and the two half sides is equal to one-half the perimeter of the original triangle
  • The area of the triangle formed by the midsegment and the two half sides is equal to one-fourth the area of the original triangle

The fact that the midsegment is half the length of the base is often called the Midsegment Theorem. You may recall that a theorem is simply a statement in mathematics that has been formally proven. Let's look at a quick example that uses the Midsegment Theorem.

In the picture below, segment DE is a midsegment of triangle ABC. If DE is parallel to AC and AC has a length of 10 feet, how long is DE?

DE is the midsegment and is parallel to AC.
Triangle with Midsegment

We know from the Midsegment Theorem that DE is one-half the length of AC. Therefore, DE must be one-half of 10 feet, or 5 feet.

Let's take a look at some examples.

Examples Using the Midsegment

In the picture below, we see that B St., Smith Rd., and Powell St. form a triangle. Jones Way forms the midsegment of this triangle. If you knew that Powell St. was 1000 meters long, you could find the length of Jones Way. Since Jones Way is the midsegment, it must be half the length of the base, Powell St.

Jones Way is the midsegment of the triangle formed by B St., Smith Rd., and Powell St.
Streets Midsegment

One-half of 1000 meters is 500 meters, therefore, Jones Way is 500 meters long.

Let's try another example.

In triangle ABC below, DE is the midsegment of the triangle. If the measure of angle BAC is 55 degrees, what is the measure of angle BDE?

In this picture, compare the measures of the left hand side angles, BAC and BDE.
Tirangle Midsegment

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Video Transcript

Definition

The midsegment of a triangle is defined as the segment formed by connecting the midpoints of any two sides of a triangle. Put simply, it divides two sides of a triangle equally. The midpoint of a side divides the side into two equal segments. As you can see in the picture below, DE is the midsegment of the triangle ABC. Point D divides segment AB into two equal parts, and point E divides segment CB into two equal parts. The side of the triangle that the midsegment does not intersect is the base of the triangle.

In this triangle, midsegment DE divides segment AB into two equal parts, and divides segment CB into two equal parts.
Triangle showing midsegment

The midsegment of a trapezoid is the segment formed by connecting the midpoints of the two legs of a trapezoid. Although the midsegment of a trapezoid is also useful in mathematics, we will not discuss it here.

Properties of the Midsegment and Midsegment Theorem

The triangle midsegment has several useful properties:

  • The midsegment is half the length of the base

The midsegment is one-half the length of the base.
Midsegment labeled with length

  • The midsegment is parallel to the base
  • The triangle formed by the midsegment and the two half sides have all the same angle measures as the original triangle
  • The perimeter of the triangle formed by the midsegment and the two half sides is equal to one-half the perimeter of the original triangle
  • The area of the triangle formed by the midsegment and the two half sides is equal to one-fourth the area of the original triangle

The fact that the midsegment is half the length of the base is often called the Midsegment Theorem. You may recall that a theorem is simply a statement in mathematics that has been formally proven. Let's look at a quick example that uses the Midsegment Theorem.

In the picture below, segment DE is a midsegment of triangle ABC. If DE is parallel to AC and AC has a length of 10 feet, how long is DE?

DE is the midsegment and is parallel to AC.
Triangle with Midsegment

We know from the Midsegment Theorem that DE is one-half the length of AC. Therefore, DE must be one-half of 10 feet, or 5 feet.

Let's take a look at some examples.

Examples Using the Midsegment

In the picture below, we see that B St., Smith Rd., and Powell St. form a triangle. Jones Way forms the midsegment of this triangle. If you knew that Powell St. was 1000 meters long, you could find the length of Jones Way. Since Jones Way is the midsegment, it must be half the length of the base, Powell St.

Jones Way is the midsegment of the triangle formed by B St., Smith Rd., and Powell St.
Streets Midsegment

One-half of 1000 meters is 500 meters, therefore, Jones Way is 500 meters long.

Let's try another example.

In triangle ABC below, DE is the midsegment of the triangle. If the measure of angle BAC is 55 degrees, what is the measure of angle BDE?

In this picture, compare the measures of the left hand side angles, BAC and BDE.
Tirangle Midsegment

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Frequently Asked Questions

Is the Midsegment half of the base?

The midsegment of a triangle is always half of the measure of the base that the segment is parallel to.

How do you find the length of a Midsegment?

If you know the length of the base of a triangle, you can find the length of the midsegment parallel to that base by taking half of the measurement.

Midsegment formula: Midsegment=1/2 the base of the triangle

How do you solve a Midsegment problem?

To solve a midsegment problem, identify the midsegment and the side of the triangle parallel to it. The midsegment = 1/2 of the parallel side. Then solve the equation to find the value.

What is special about a Midsegment?

A midsegment connects the midpoints of two sides of a triangle making it 1/2 of the length of the third side. The midsegment and the third side will always be parallel.

What is the formula to find the Midsegment of a triangle?

If you know the length of the base of a triangle, you can find the length of the midsegment parallel to that base by taking half of the measurement.

Midsegment formula: Midsegment=1/2 the base of the triangle

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