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Midsegment: Theorem & Formula

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  • 0:04 Definition
  • 0:56 Properties of the…
  • 2:08 Examples Using the Midsegment
  • 3:35 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Often

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

Learn about the midsegment of a triangle and how it's found. Read about the properties of the midsegment as they relate to the base, angles, perimeter and area, and then test your knowledge with a short quiz.

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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