# Midsegment of a triangle

🏆Practice middle segment of a triangle

The midsegment of a triangle has three main properties:

• The midsegment crosses exactly through the middle of the two sides that determine it.
• The midsegment is parallel to the third side of the triangle.
• The midsegment measures half the length of the side arranged parallel to it.

Let's look at the properties of the midsegment of a triangle in the following illustration:

If $AD=CD$
$AE=BE$

then $2DE=CB$
$DE∥CB$

## Test yourself on middle segment of a triangle!

Given that DE is a middle section in triangle ABC, what is the length of side DE?

## Proof of the Midsegment of a Triangle

We can demonstrate that there is a midsegment in a triangle if at least one of the following conditions is met:

• If in a triangle there is a straight line that extends from the midpoint of one side to the midpoint of another side, we can determine that it is a midsegment and, therefore, that it measures half the length of the third side, to which, in fact, it is also parallel.
• If a straight line cuts one of the sides of a triangle and it is parallel to another side of the triangle, it means that it is a midsegment and that, therefore, it also cuts the third side of the triangle and measures half the length of the side that is parallel to it.
• If in a triangle there is a segment whose ends are located on two of its sides, measures half the length of the third side and is parallel to it, we can determine that said segment is a midsegment and, therefore, cuts the sides it touches right in the middle.

## Let's look at an example

Given $⊿ABC$

$DE∥AB$
$AD=CD$

To prove:
$DE=2AB$

Solution:
If a straight line cuts one of the sides of a triangle – given that$DE$ cuts the edge $AC$,
and is parallel to another side of the triangle,

Given that: $DE∥AB$
it means that it is a midsegment and therefore, measures half the length of the side it is parallel to.

That is:
$DE=2AB$

If you are interested in this article, you might also be interested in the following articles:

• Midsegment
• Midsegment of a trapezoid
• Sum of the interior angles of a polygon
• Angles in regular hexagons and octagons
• Measure of an angle of a regular polygon
• Sum of the exterior angles of a polygon
• Relationships between angles and sides of a triangle
• The relationship between the lengths of the sides of a triangle
• Identification of an isosceles triangle
• How is the area of a trapezoid calculated?
• Symmetry in trapezoids
• Diagonals of an isosceles trapezoid
• How is the perimeter of a trapezoid calculated?
• Characteristics and types of trapezoids

In the Tutorela blog, you will find a variety of articles on mathematics.

## Examples and exercises with solutions of the midsegment of a triangle

### Exercise #1

Given that DE is a middle section in triangle ABC, what is the length of side DE?

4

### Exercise #2

Given that DE is a middle section in triangle ABC, what is the length of side DE?

5

### Exercise #3

Given that DE is a middle section in triangle ABC, what is the length of side DE?

4.5

### Exercise #4

Given that DE is a middle section in triangle ABC, what is the length of side DE?

9

### Exercise #5

Given that DE is a middle section in triangle ABC, what is the length of side DE?