Midsegment 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 $2DE=CB$
$DE∥CB$

Detailed explanation

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Test yourself on midsegment of a triangle!

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

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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

We can demonstrate that there is a midsegment in a triangle

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:
$DE=2AB$

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Examples and exercises with solutions of the midsegment of a triangle

Exercise #1

Calculate the perimeter of triangle ADE given that DE is the midsegment of triangle ABC.

Video Solution

Step-by-Step Solution

In order to calculate the perimeter of triangle $\triangle ADE$ we need to find the lengths of its sides,

Let's now refer to the given information that $DE$ is a median in $\triangle ABC$ and therefore a median in a triangle equals half the length of the side it does not intersect, additionally we'll remember the definition of a median in a triangle as a line segment that extends from the midpoint of one side to the midpoint of another side, we'll write the property mentioned (a) and the fact derived from the given definition (b+c):