What if you were given a triangle with a line segment drawn through it from one side to the other? How could you use information about the triangle's side lengths to determine if that line segment is parallel to the third side? After completing this Concept, you'll be able to answer questions like this one.
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CK-12 Foundation: Chapter7TriangleProportionalityA
James Sousa: Triangle Proportionality Theorem
James Sousa: Using the Triangle Proportionality Theorem to Solve for Unknown Values
Guidance
Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two sides into congruent halves. The midsegment divides those two sides proportionally.
Investigation: Triangle Proportionality
Tools Needed: pencil, paper, ruler
- Draw . Label the vertices.
- Draw so that is on and is on . and can be anywhere on these sides.
- Is ? Why or why not? Measure and . Then set up the ratios and . Are they equal?
- Draw a second triangle, . Label the vertices.
- Draw so that is on and is on AND .
- Is ? Why or why not? Measure and . Then set up the ratios and . Are they equal?
From this investigation, it is clear that if the line segments are parallel, then divides the sides proportionally.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Proof of the Triangle Proportionality Theorem:
Given : with
Prove :
Statement | Reason |
---|---|
1. | Given |
2. | Corresponding Angles Postulate |
3. | AA Similarity Postulate |
4. | Segment Addition Postulate |
5. | Corresponding sides in similar triangles are proportional |
6. | Substitution PoE |
7. | Separate the fractions |
8. | Substitution PoE (something over itself always equals 1) |
9. | Subtraction PoE |
Example A
A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?
The midsegment’s endpoints are the midpoints of the two sides it connects. The midpoints split the sides evenly. Therefore, the ratio would be or . Both of these reduce to 1:1.
Example B
In the diagram below, . Find .
Use the Triangle Proportionality Theorem.
Example C
Is ?
Use the Triangle Proportionality Converse. If the ratios are equal, then the lines are parallel.
and
Because the ratios are equal, .
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7TriangleProportionalityB
Vocabulary
A line segment that connects two midpoints of the sides of a triangle is called a midsegment . A midpoint is a point that divides a segment into two equal pieces. Pairs of numbers are proportional if they are in the same ratio.
Guided Practice
Use the diagram to answers questions 1-5. .
1. Name the similar triangles. Write the similarity statement.
2.
3.
4.
5.
Answers:
1.
2. DF
3. DC
4. FE
5. DF; DB
Practice
Use the diagram to answer questions 1-5. .
- Find .
- Find .
- Find .
- Find .
- What is ?
- What is ?
- We know that and . Why is ?
Use the given lengths to determine if .
Find the unknown length.
- What is the ratio that the midsegment divides the sides into?