Midsegment

The midsegment is a segment that connects the midpoints of 2 sides.
It's very simple to remember the meaning of this term since the word "middle" already tells us that it is about the midpoint, so when we come across the concept of "midsegment" we'll remember that it connects the midpoints of two sides.
We're here to teach you everything you need to know about the midsegment, from the proof to the wonderful properties of the segment that will help us solve exercises.
First, we'll talk about the midsegment of a triangle and then we'll move on to the midsegment of a trapezoid.

Midsegment of a Triangle

The midsegment of a triangle crosses the middle of two sides, is parallel to the third side
and is also half its length.

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

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

Midsegment of a Trapezoid

The midsegment of a trapezoid divides the two sides it originates from into two equal parts, and is also parallel to both bases of the trapezoid and measures half the length of these bases.

We can demonstrate that there is a midsegment in a trapezoid provided that, at least, one of the following conditions is met:

If there is a straight line in a trapezoid that extends from the midpoint of one side to the midpoint of another side, we can determine that it is a midsegment. As a result, it is parallel to both bases of the trapezoid and its length is half that of these bases.
If there is a straight line that extends from one side of a trapezoid and is parallel to one of the trapezoid's bases, we can confirm that it is a midsegment. Therefore, it is parallel to both bases of the trapezoid, its length is half that of these two bases, and it also bisects the second side that it touches.

Detailed explanation

Practice Midsegment

Question 1

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

Question 2

Given an isosceles trapezoid, is the dashed segment a middle segment of the trapezoid?

Question 3

Is the dashed segment the midsegment of the isosceles trapezoid below?

Question 4

In which figure is the dotted line the midsegment in the trapezoid?

Question 5

In which figure is the dashed line the midsection of the trapezoid?

Examples with solutions for Midsegment

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):

$DE=\frac{1}{2}BC$ b.

$AD=\frac{1}{2}AB$ c.

$AE=\frac{1}{2}AC\\$ Additionally, the given data in the drawing are:

$BC=8$ e.

$AB=6$ f.

$AC=10$ Therefore, we will substitute d', e', and f' respectively in a', b', and c', and we get:

$DE=\frac{1}{2}BC=\frac{1}{2}\cdot8=4$ h.

$AD=\frac{1}{2}AB=\frac{1}{2}\cdot6=3$ i.

$AE=\frac{1}{2}AC=\frac{1}{2}\cdot10=5$

Therefore the perimeter of $\triangle ADE$ is:

$P_{ADE}=DE+AD+AE=4+3+5=12$ Therefore the correct answer is answer d.

Answer

Exercise #2

Given an isosceles trapezoid, is the dashed segment a middle segment of the trapezoid?

Video Solution

Answer

Answer

Question 1

Below is an isosceles trapezium.

EF is parallel to the base of the trapezium.

True or false: EF is the midsection of the trapezoid.

Question 2

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

Question 3

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

Question 4

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

Question 5

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

Exercise #6

Below is an isosceles trapezium.

EF is parallel to the base of the trapezium.

True or false: EF is the midsection of the trapezoid.

Video Solution

Answer

Answer

4.5

Exercise #10

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

Video Solution

Answer

Question 1

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

Question 2

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

Question 3

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

Question 4

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

Question 5

Answer

7.5