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Triangle Midsegment Theorem

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Triangle Midsegment Theorem

Study Guide

Key Definition

In a triangle, a midsegment connecting the midpoints of two sides is parallel to the third side and is half as long as that side: $\overline{DE} \parallel \overline{BC}$ and $DE = \frac{1}{2}BC$

Important Notes

A midsegment connects the midpoints of two sides of a triangle.
$\overline{DE} \parallel \overline{BC}$ implies similar triangles.
The midsegment theorem is a powerful tool in geometry.
Midsegments divide triangles into smaller, similar triangles.
Useful for calculating unknown sides using proportional relationships.

Mathematical Notation

$\angle$ means angle

$\triangle$ means triangle

$\parallel$ means parallel

$\overline{AB}$ means line segment AB

$\frac{1}{2}$ represents half or one-half

$D is the midpoint of AB$

$E is the midpoint of AC$

Remember to use proper notation when solving problems

Why It Works

Since DE is parallel to BC, angles ∠ADE and ∠ABC are congruent and angles ∠AED and ∠ACB are congruent. Therefore, triangles ADE and ABC are similar by the AA criterion. From the similarity, corresponding sides satisfy DE/BC = AD/AB = 1/2, so DE = 1/2 BC, proving the midsegment theorem.

Remember

The midsegment is always parallel to the third side and is half its length: $DE = \frac{1}{2}BC$

Quick Reference

Midsegment Property:$DE = \frac{1}{2}BC$

Understanding Triangle Midsegment Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In triangle ABC, let D and E be the midpoints of sides AB and AC respectively. The segment DE joining these midpoints is called a midsegment. The midsegment theorem states that DE is parallel to BC and its length is half of BC, so DE = 1/2 BC.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In triangle $\triangle ABC$, if $DE$ is a midsegment, what is the length of $DE$ if $BC = 10$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Alex wants to create a triangle garden and uses the midsegment theorem to ensure symmetry. If side $\overline{BC}$, the longest side of the triangle, measures $16$ feet, what should be the length of the midsegment $DE$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given $\triangle ABC$ with midpoints $D$ and $E$, determine the relationship between $\overline{DE}$ and $\overline{BC}$.

Challenge Quiz

Single Choice Quiz

Advanced

For $\triangle ABC$ with $DE$ as a midsegment, if $AD = 4$ and $DB = 4$, what is $AB$?

Recap

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Review key concepts and takeaways