A line parallel to one side of a triangle cuts the other two sides in the same ratio.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it is true
The parallel makes equal corresponding angles, so the small triangle is similar to the whole.
The midsegment
Joining the midpoints of two sides gives a segment parallel to the third and exactly half its length.
Using it
Set the two ratios equal and solve for the unknown piece.
A parallel cut divides proportionally
A line parallel to one side of a triangle cuts the other two sides in the same ratio. It follows directly from AA similarity, since the parallel line creates a smaller triangle with the same angles.
The converse proves parallelism
If a line divides two sides of a triangle proportionally, it is parallel to the third side. This direction is how you prove lines parallel from measurements alone, and it is a genuinely useful tool.
Which ratio to write
The theorem compares part to part, or part to whole, but you must be consistent on both sides. Writing top-to-bottom for one side and whole-to-part for the other is the classic mistake in these problems.
Where it appears
Dividing a segment into equal parts with only a straightedge and compass, and the midsegment theorem — that the segment joining two midpoints is parallel to and half the third side — are both consequences of this one result.
Step 2: Try It Yourself
Tap and try it out.
Before: 10
After: 5
The factor is less than 1, so the bar got shorter. Multiplying does not always make things bigger.
Step 3: Watch an Example
One step at a time.
Watch Kofi Use the Ratios
A parallel cut divides one side into 3 and 6, and the other into 4 and x.
- Step 1
The ratios must match: 3 to 6 equals 4 to x.
Step 4: Your Turn
Practice makes it stick.
The Midsegment
Problem 1 of 2
The third side is 14. How long is the midsegment?
The Ratio
Problem 2 of 2
A cut divides one side 2 and 4, the other 3 and x. What is x?
Parallel Cuts
1 of 8
Third side 20. Midsegment?
2 of 8
Midsegment 6. Third side?
3 of 8
Cut divides 4 and 8, other side 5 and x. What is x?
4 of 8
Cut divides 6 and 6, other side 9 and x. What is x?
5 of 8
Is a midsegment parallel to the third side? 1 for yes, 0 for no.
6 of 8
Third side 9. Midsegment, as a decimal?
7 of 8
Cut divides 2 and 10, other side 1 and x. What is x?
8 of 8
Does the parallel line create a similar triangle? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Third side 16. Midsegment?
What You Learned
- A line parallel to one side cuts the other two in the same ratio.
- It works because the parallel creates a similar triangle.
- A midsegment joins two midpoints and is half the third side.