MATHEMATICS • GEOMETRY • TRIANGLES
What Is the Midpoint Theorem and Its Converse?
The midpoint theorem connects the midpoints of two sides of a triangle. Its converse tells us what happens when a line passes through the midpoint of one side and is parallel to another side.
Both results are useful for proving lines parallel, finding unknown lengths and solving geometry questions.
1. What Is the Midpoint of a Line Segment?
The midpoint of a line segment is the point that divides the segment into two equal parts. If M is the midpoint of AB, then AM = MB.
If M is the midpoint of AB:
AM = MB
For example, if AB = 12 cm and M is its midpoint, then AM = MB = 6 cm.
2. Statement of the Midpoint Theorem
The midpoint theorem states: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of its length.
In triangle ABC, let D be the midpoint of AB and E be the midpoint of AC. Then:
DE ∥ BC
DE = ½ BC
3. Proof of the Midpoint Theorem
Given: In triangle ABC, D is the midpoint of AB and E is the midpoint of AC.
To prove: DE ∥ BC and DE = ½ BC.
Proof:
- Since D is the midpoint of AB, AD = DB. Therefore, AB/AD = 2.
- Since E is the midpoint of AC, AE = EC. Therefore, AC/AE = 2.
- Thus, AB/AD = AC/AE.
- By the converse of the Basic Proportionality Theorem (BPT), DE ∥ BC.
- Since DE ∥ BC, triangles ADE and ABC are similar.
- Therefore, AD/AB = DE/BC. As AD/AB = 1/2, we get DE/BC = 1/2.
Hence, DE ∥ BC and DE = ½ BC. Proved.
4. Statement of the Converse of the Midpoint Theorem
The converse states: A line drawn through the midpoint of one side of a triangle, parallel to another side, bisects the third side.
In triangle ABC, if D is the midpoint of AB and a line through D parallel to BC meets AC at E, then E is the midpoint of AC.
If AD = DB and DE ∥ BC, then
AE = EC
5. Proof of the Converse
Given: In triangle ABC, D is the midpoint of AB and DE ∥ BC, where E lies on AC.
To prove: E is the midpoint of AC, that is, AE = EC.
Proof:
- Since DE ∥ BC, by the Basic Proportionality Theorem, AD/DB = AE/EC.
- D is the midpoint of AB, so AD = DB.
- Therefore, AD/DB = 1, which gives AE/EC = 1.
- Hence, AE = EC.
Therefore, E is the midpoint of AC. Proved.
6. Solved Examples
Example 1: Find the length of the midpoint segment
In triangle ABC, D and E are the midpoints of AB and AC. If BC = 18 cm, find DE.
Solution: By the midpoint theorem, DE = ½ BC.
DE = ½ × 18 = 9 cm.
Answer: 9 cm.
Example 2: Find the third side
D and E are the midpoints of two sides of a triangle. If DE = 7.5 cm and DE ∥ BC, find BC.
Solution: DE = ½ BC.
BC = 2 × DE = 2 × 7.5 = 15 cm.
Answer: 15 cm.
Example 3: Use the converse
In triangle PQR, S is the midpoint of PQ. A line through S parallel to QR meets PR at T. If PT = 6 cm, find TR.
Solution: By the converse of the midpoint theorem, T is the midpoint of PR. Therefore, PT = TR.
So, TR = 6 cm.
Answer: 6 cm.
Example 4: Find an unknown side
In triangle ABC, D and E are midpoints of AB and AC. If DE = 3x − 1 and BC = 20 cm, find x.
Solution: DE = ½ BC = 10 cm.
3x − 1 = 10, so 3x = 11 and x = 11/3.
Answer: x = 11/3.
7. Midpoint Theorem vs Its Converse
| Midpoint theorem | Converse of midpoint theorem |
|---|---|
| Starts with the midpoints of two sides. | Starts with the midpoint of one side and a parallel line. |
| Proves the joining segment is parallel to the third side. | Proves the parallel line bisects the third side. |
| Also proves the joining segment is half the third side. | Its main conclusion is that the third side is divided into two equal parts. |
8. Common Mistakes to Avoid
- Do not confuse the midpoint theorem with its converse; their given information and conclusions are different.
- Use the midpoint theorem to write DE = ½ BC, not DE = 2BC.
- For the converse, make sure the line is parallel to the specified side and passes through the midpoint of another side.
- A midpoint divides a segment into two equal lengths; it does not necessarily divide an angle into equal angles.
9. Practice Questions
- D and E are the midpoints of AB and AC in triangle ABC. If BC = 24 cm, find DE.
- If the segment joining the midpoints of two sides of a triangle is 8.5 cm, find the third side.
- In triangle XYZ, M is the midpoint of XY. A line through M parallel to YZ meets XZ at N. If XN = 4.2 cm, find NZ.
- In triangle ABC, D is the midpoint of AB and DE ∥ BC. What can you conclude about E on AC?
- True or false: The segment joining the midpoints of two sides of a triangle is equal in length to the third side.
Answers
- DE = ½ × 24 = 12 cm.
- The third side = 2 × 8.5 = 17 cm.
- By the converse, N is the midpoint of XZ, so NZ = XN = 4.2 cm.
- E is the midpoint of AC, so AE = EC.
- False. It is half the length of the third side.
Frequently Asked Questions
What is the formula for the midpoint theorem?
If D and E are the midpoints of two sides of triangle ABC, then DE ∥ BC and DE = ½ BC.
What does the converse of the midpoint theorem prove?
It proves that a line through the midpoint of one side of a triangle, parallel to a second side, bisects the third side.
Can the midpoint theorem be used to find missing lengths?
Yes. The segment joining two midpoints is half the third side, so you can multiply or divide by 2 to find an unknown length.
Which theorem is used in the proofs?
The Basic Proportionality Theorem (BPT) and the similarity of triangles are commonly used to prove the midpoint theorem and its converse.