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High School Geometry: Help and Review13 chapters | 162 lessons

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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.

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.

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.

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.

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.

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High School Geometry: Help and Review13 chapters | 162 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Constructing the Median of a Triangle 4:47
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- How to Find the Height of a Triangle 4:41
- Hypotenuse: Definition & Formula 4:38
- Included Angle of a Triangle: Definition & Overview 4:06
- Inscribed Angle: Definition, Theorem & Formula 3:40
- Interior Angle Theorem: Definition & Formula 4:37
- Median of a Triangle: Definition & Formula 3:12
- Midsegment: Theorem & Formula 4:18
- Go to Properties of Triangles: Help and Review

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