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

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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?

Video Solution

Answer

4

Exercise #2

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

Video Solution

Answer

5

Exercise #3

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

Video Solution

Answer

4.5

Exercise #4

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

Video Solution

Answer

9

Exercise #5

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

Video Solution

Answer

11

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Question 1

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