# All about the Midsegment of a Triangle

## 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.

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.

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.

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.

## 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:

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 2**

**Practice Problem 3**

#### 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

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.

#### Finding the Midpoint of a side of a triangle

1. Using a straightedge, draw a triangle of any type.

2. With the needle edge of the compass on one of the vertices of the triangle and the hinge open, the pencil end is farther than half of the segment for which we are finding the midpoint to make a semicircle.

3. Repeat this process with the other vertex for the segment which we are finding the midpoint

4. Using a straight edge, line up the straightedge with the intersection points of the two semicircles. Mark the targeted segment and label it M. This is the midpoint of one side of the triangle.

#### Find the Midpoint of a Second Side

1. Follow steps 2-4.

2. Label this new midpoint N.

#### Draw the Segment MN

1. Use a straightedge.

2. At this point, we have constructed the midsegment of the triangle.

3. Now we need to check the measure of the midsegment of the triangle and compare it to the length of the third side of the triangle.

#### Check that MN is 1/2 the Length

Check that MN is 1/2 the length of the side of the triangle that it is parallel to.

1. Using the needlepoint of the compass and the pencil end of the compass, measure the length of MN.

2. Without disturbing the compass measurement, place the needle side of the compass on the vertex. This will be the endpoint of the segment of the side parallel to the midsegment of the triangle. Make a mark on the third side of the triangle.

3. Place the needle side on the new mark. The pencil end should line up with the other endpoint of the third side of the triangle, proving the measure of the midsegment of a triangle fits in its parallel segment two times (or the midsegment of a triangle is 1/2 of the length of the segment it is parallel to).

## Lesson Summary

A **midsegment** of a triangle is found by connecting the midpoints of two sides of a triangle. The midsegment of a triangle will be parallel to the third side of the triangle. Since there are three sides to a triangle, there are three Midsegments in every triangle. The value of the midsegment of a triangle is half of the value of the side of which it is parallel.

## 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 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*?

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.

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*?

The third property of the midsegment tells us that the midsegment forms a second, smaller triangle that has all the same angle measures as the original triangle. This means that if angle *BAC* has a measure of 55 degrees, angle *BDE* must also have a measure of 55 degrees.

The area relationship is frequently seen in the **Sierpinski triangle**. This special design is made by drawing a triangle and then drawing the three midsegments. The new central triangle created (shown in red in the left side triangle below) has an area equal to one-fourth the area of the original triangle. The process of drawing the midsegments of unshaded triangles, and shading the new central triangle can be repeated to create the design, as shown in the right side triangle below.

## Lesson Summary

The **midsegment of a triangle** is defined as the segment formed by connecting the midpoints of any two sides of a triangle. It has the following properties:

1) It is half the length of the base of the triangle.

2) It is parallel to the base.

3) It forms a smaller triangle that has all the same angle measures as the original triangle.

4) It forms a smaller triangle with a perimeter equal to one-half the perimeter of the original triangle.

5) It forms a smaller triangle with an area equal to one-fourth the area of the original triangle.

## Properties of the Midsegment of a Triangle

- It connects the two midpoints of the two sides of a triangle.
- It is equal to one half the length of the base.
- It is parallel to the base.
- It forms a smaller triangle with all the same angle measures, one-half the perimeter, and one-fourth the area of the original triangle.

## Learning Outcomes

Studying this information on the midsegment could enable you to do these things:

- Note the definition and purpose of the midsegment of a triangle
- State the properties of a triangle midsegment
- Use the Midsegment Theorem

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Create your account

## 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.

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 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*?

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.

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*?

The third property of the midsegment tells us that the midsegment forms a second, smaller triangle that has all the same angle measures as the original triangle. This means that if angle *BAC* has a measure of 55 degrees, angle *BDE* must also have a measure of 55 degrees.

The area relationship is frequently seen in the **Sierpinski triangle**. This special design is made by drawing a triangle and then drawing the three midsegments. The new central triangle created (shown in red in the left side triangle below) has an area equal to one-fourth the area of the original triangle. The process of drawing the midsegments of unshaded triangles, and shading the new central triangle can be repeated to create the design, as shown in the right side triangle below.

## Lesson Summary

The **midsegment of a triangle** is defined as the segment formed by connecting the midpoints of any two sides of a triangle. It has the following properties:

1) It is half the length of the base of the triangle.

2) It is parallel to the base.

3) It forms a smaller triangle that has all the same angle measures as the original triangle.

4) It forms a smaller triangle with a perimeter equal to one-half the perimeter of the original triangle.

5) It forms a smaller triangle with an area equal to one-fourth the area of the original triangle.

## Properties of the Midsegment of a Triangle

- It connects the two midpoints of the two sides of a triangle.
- It is equal to one half the length of the base.
- It is parallel to the base.
- It forms a smaller triangle with all the same angle measures, one-half the perimeter, and one-fourth the area of the original triangle.

## Learning Outcomes

Studying this information on the midsegment could enable you to do these things:

- Note the definition and purpose of the midsegment of a triangle
- State the properties of a triangle midsegment
- Use the Midsegment Theorem

To unlock this lesson you must be a Study.com Member.

Create your account

#### 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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