Midsegment of a Triangle - Examples, Exercises and Solutions

Understanding Midsegment of a Triangle

Complete explanation with examples

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

Detailed explanation

Practice Midsegment of a Triangle

Test your knowledge with 6 quizzes

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

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Examples with solutions for Midsegment of a Triangle

Step-by-step solutions included
Exercise #1

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

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Step-by-Step Solution

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

Let's now refer to the given information that DE DE is a median in ABC \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):

a.

DE=12BC DE=\frac{1}{2}BC b.

AD=12AB AD=\frac{1}{2}AB c.

AE=12AC AE=\frac{1}{2}AC\\ Additionally, the given data in the drawing are:

d.

BC=8 BC=8 e.

AB=6 AB=6 f.

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

g.

DE=12BC=128=4 DE=\frac{1}{2}BC=\frac{1}{2}\cdot8=4 h.

AD=12AB=126=3 AD=\frac{1}{2}AB=\frac{1}{2}\cdot6=3 i.

AE=12AC=1210=5 AE=\frac{1}{2}AC=\frac{1}{2}\cdot10=5 888444333333555555AAABBBCCCDDDEEE

Therefore the perimeter of ADE \triangle ADE is:

j.

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

Answer:

12

Video Solution
Exercise #2

ABC is a right triangle.

DE is parallel to BC and is the midsection of triangle ABC.

Given in cm:

BC = 5

AC = 13

Calculate the area of the trapezoid DECB.

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Step-by-Step Solution

We are tasked with finding the area of trapezoid DECB in the right triangle ABC where DE is the midsegment parallel to BC. Given BC=5 BC = 5 cm, AC=13 AC = 13 cm, let us calculate the area step-by-step.

  • First, use the Pythagorean theorem to find the length of AB AB . Since triangle ABC is a right triangle at B B , we have:

AB=AC2BC2=13252=16925=144=12 AB = \sqrt{AC^2 - BC^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 cm.

  • The midsegment DE of triangle ABC, being parallel to the base BC, is half the length of BC:

DE=12×BC=12×5=2.5 DE = \frac{1}{2} \times BC = \frac{1}{2} \times 5 = 2.5 cm.

  • Now, to find the area of trapezoid DECB, which has bases DE and BC, the height is the same as AB AB (the vertical side of triangle ABC):

A=12×(b1+b2)×h=12×(2.5+5)×12 A = \frac{1}{2} \times (b_1 + b_2) \times h = \frac{1}{2} \times (2.5 + 5) \times 12 .

Calculate the expression:

  • A=12×7.5×12=12×90=45 A = \frac{1}{2} \times 7.5 \times 12 = \frac{1}{2} \times 90 = 45 cm².
  • Since DECB is half of the total triangle, the area is half of 45.

So, the area of trapezoid DECB is 45÷2=22.5 45 \div 2 = 22.5 cm².

The solution to the problem is 22.5 \boxed{22.5} .

Answer:

22.5

Video Solution
Exercise #3

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

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Step-by-Step Solution

Answer:

4

Video Solution
Exercise #4

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

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Step-by-Step Solution

Answer:

5

Video Solution
Exercise #5

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

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Step-by-Step Solution

Answer:

4.5

Video Solution

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