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Triangle Proportionality Theorem

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Triangle Proportionality Theorem

Study Guide

Key Definition

If a line parallel to one side of a <a href="triangles">$\triangle$</a> intersects the other two sides, it divides those sides proportionally. For example, if $\overline{DE} \parallel \overline{BC}$, then $\frac{AD}{DB} = \frac{AE}{EC}$.

Important Notes

The theorem applies only if the line is parallel to one side of the triangle.
Proportionality means that the ratios of the segments are equal.
This theorem can help solve for unknown lengths in geometric problems.
The (perpendicular) distance between two parallel lines is the same everywhere.
Ensure calculations use correct segment lengths.

Mathematical Notation

$\parallel$ means parallel lines

$\frac{a}{b}$ represents a fraction

$=$ represents equality

$\triangle$ represents a triangle

$\angle$ denotes an angle

Remember to use proper notation when solving problems

Why It Works

This theorem works because of the properties of parallel lines and similar triangles. If two triangles are similar, their corresponding sides are proportional.

Remember

Always check that the line is parallel to a side of the triangle before using the proportionality theorem.

Quick Reference

Triangle Proportionality Theorem:$\frac{AD}{DB} = \frac{AE}{EC}$

Understanding Triangle Proportionality Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When a line is parallel to one side of a <a href="triangles">$\triangle$</a>, it divides the other two sides proportionally: $\frac{AD}{DB} = \frac{AE}{EC}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $\overline{DE} \parallel \overline{BC}$ in <a href="triangles">$\triangle ABC$</a>, and $AD = 3$, $DB = 2$, $AE = 4.5$, find $EC$.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is in the shape of triangle ABC with BC as the base. A rail DE is drawn parallel to BC, intersecting AB at D and AC at E. If AD = 4 m, DB = 6 m, and AE = 3 m, find the length of EC using $\frac{AD}{DB} = \frac{AE}{EC}$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a <a href="triangles">$\triangle$</a> with sides divided by a parallel line, prove the segments are proportional.

Challenge Quiz

Single Choice Quiz

Advanced

If $\overline{XY} \parallel \overline{BC}$ in <a href="triangles">$\triangle ABC$</a>, and $AX = 5$, $XB = 4$, $AY = 7.5$, find the length of $YC$.

Recap

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Review key concepts and takeaways