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:

Midsegment of a triangle

If AD=CD AD=CD
AE=BEAE=BE

then 2DE=CB2DE=CB
DECBDE∥CB

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Test yourself on middle segment of a triangle!

einstein

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

666888101010AAABBBCCCEEEDDD

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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⊿ABC

DEABDE∥AB
AD=CDAD=CD

To prove:
DE=2ABDE=2AB

Solution:
If a straight line cuts one of the sides of a triangle – given thatDEDE cuts the edge ACAC,
and is parallel to another side of the triangle,

Given that: DEAB 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=2ABDE=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?

666888101010AAABBBCCCEEEDDD

Video Solution

Answer

4

Exercise #2

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

444666101010AAABBBCCCDDDEEE

Video Solution

Answer

5

Exercise #3

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

121212999121212AAABBBCCCDDDEEE

Video Solution

Answer

4.5

Exercise #4

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

181818121212181818AAABBBCCCDDDEEE

Video Solution

Answer

9

Exercise #5

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

222222666222222AAABBBCCCEEEDDD

Video Solution

Answer

11

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