Midsegment Practice Problems - Triangle & Trapezoid Worksheets

Master midsegment theorems with step-by-step practice problems. Learn properties of triangle and trapezoid midsegments through interactive exercises and solutions.

📚Master Midsegment Properties Through Guided Practice
  • Identify midsegments in triangles using midpoint conditions
  • Apply the triangle midsegment theorem to find parallel segments
  • Calculate lengths using the half-length property of midsegments
  • Prove midsegment properties in trapezoids step-by-step
  • Solve for unknown measurements in midsegment problems
  • Connect midpoints to create parallel segments confidently

Understanding Midsegment

Complete explanation with examples

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:

  1. 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.
  2. 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.
  3. 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 Triangle

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:

  1. 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.
  2. 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.

Midsegment of a Trapezoid

Detailed explanation

Practice Midsegment

Test your knowledge with 9 quizzes

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

121212141414111111AAABBBCCCDDDEEE

Examples with solutions for Midsegment

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=12⋅8=4 DE=\frac{1}{2}BC=\frac{1}{2}\cdot8=4 h.

AD=12AB=12⋅6=3 AD=\frac{1}{2}AB=\frac{1}{2}\cdot6=3 i.

AE=12AC=12⋅10=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.

555131313AAABBBCCCDDDEEE

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=AC2−BC2=132−52=169−25=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

Below is an isosceles trapezium.

EF is parallel to the base of the trapezium.

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

AAABBBDDDCCCEEEFFF

Step-by-Step Solution

Answer:

True

Video Solution
Exercise #4

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

Step-by-Step Solution

Answer:

Not true

Video Solution
Exercise #5

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

181818121212181818AAABBBCCCDDDEEE

Step-by-Step Solution

Answer:

9

Video Solution

Frequently Asked Questions

Everything you need to know about Midsegment

What is a midsegment in geometry?

+
A midsegment is a line segment that connects the midpoints of two sides of a triangle or trapezoid. In triangles, it's parallel to the third side and half its length. In trapezoids, it's parallel to both bases and equals half their combined length.

How do you prove a segment is a midsegment of a triangle?

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You can prove a midsegment exists if: 1) It connects midpoints of two sides, 2) It's parallel to the third side and cuts another side in half, or 3) It's half the length of the third side and parallel to it.

What are the key properties of triangle midsegments?

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Triangle midsegments have two main properties: they are parallel to the third side of the triangle, and they measure exactly half the length of that third side. These properties make solving geometry problems much easier.

How is a trapezoid midsegment different from a triangle midsegment?

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A trapezoid midsegment connects midpoints of the non-parallel sides (legs). It's parallel to both bases and its length equals half the sum of both base lengths, not just half of one side like in triangles.

What formula do you use for trapezoid midsegment length?

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The trapezoid midsegment length formula is: EF = (AB + DC)/2, where EF is the midsegment and AB, DC are the parallel bases. This means the midsegment equals the average of the two base lengths.

Why are midsegments always parallel to triangle sides?

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Midsegments are parallel due to the Triangle Midsegment Theorem. When you connect midpoints of two triangle sides, you create similar triangles with a 1:2 ratio, which mathematically guarantees the midsegment is parallel to the third side.

How do you find missing measurements using midsegments?

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Use these relationships: In triangles, if the midsegment = x, then the parallel side = 2x. In trapezoids, if midsegment = m and one base = a, then the other base = 2m - a. Always apply the parallel and half-length properties.

What are common mistakes when working with midsegments?

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Common errors include: forgetting the midsegment is half the length (not double), confusing which sides are parallel, not verifying midpoint conditions, and mixing up triangle vs trapezoid formulas. Always check that endpoints are true midpoints.

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