Midpoint Theorem
Midpoint theorem (proof, converse, & examples)
Arthur C. Clarke, renowned author, famously said, "Any sufficiently advanced technology is indistinguishable from magic," and that can include the advanced understanding of mathematics. The Midpoint Theorem may seem almost like an illusion, but it is both real and remarkably useful in geometry.
What is a midpoint?
The midpoint of a line segment, such as the line segment forming a side of a triangle, is the single point equidistant from both endpoints of the line segment.
| Concept | Description |
|---|---|
| Midpoint | The unique point on a line segment that divides it into two equal parts |
| Key Property | Every line segment has exactly one midpoint |
| Notation | If M is the midpoint of segment AB, then AM = MB |
Draw or imagine a line segment SK, and between the endpoints S and K are three other points, N, A, and C. Which point is the midpoint of line segment SK?
Point A is halfway between Points S and K. For any line segment, only one midpoint exists.
The midpoint theorem
The midpoint theorem states that the line segment joining the midpoints of two sides of any triangle is parallel to the third side and half its length.
| Assertion | Statement |
|---|---|
| Assertion 1 (Parallelism) | The segment connecting the midpoints of two sides is parallel to the third side |
| Assertion 2 (Length) | That segment is exactly half the length of the third side |
This at first sounds like nothing but brave talk, so let's test it. The Theorem has two assertions. The first is that, for any triangle, connecting midpoints of two sides will produce a line segment parallel to the third side.
Draw △NET with side ET = 38 meters.
If we find the midpoint of NE (call it Point M) and the midpoint of TN (Point A), the Midpoint Theorem states that line segment MA is parallel to the triangle's side ET.
The other part of the Midpoint Theorem says that new line segment, MA, is exactly half the distance of the third side, ET. So if ET is 38 meters, how long is MA?
We hope you said 19 meters!
We can add in the midpoint of ET and call it Point G.
Now we can strike a line segment, MG, parallel to TN and half its length. If we tell you TN is 32 meters, what is MG? You should say 16 meters!
With three midpoints, we can construct the third line segment, AG ∥ NE. Knowing AG = 12.5 meters, what is the length of original triangle side NE? Sure, it is 25 meters!
You have even more power in the Midpoint Theorem than you perhaps realize. Since you have struck the midpoints of each side, you know all these distances:
- NM
- ME
- EG
- GT
- TA
- AN
Midpoint theorem converse
The midpoint theorem works conversely, too: if you draw a line parallel to a side of a triangle through one side's midpoint, it will automatically intersect the midpoint of the remaining side.
| Property | Description |
|---|---|
| Converse Statement | A line drawn through the midpoint of one side, parallel to a second side, bisects the third side |
| Similar Triangles | The triangle formed by connecting midpoints is similar to the original triangle (all corresponding angles are equal) |
| Congruent Sub-Triangles | The four smaller triangles created by joining all three midpoints are congruent to each other |
| Area Relationship | Each of the four congruent sub-triangles has exactly one-quarter the area of the original triangle |
Want another bit of mathematical "magic?" The triangle created by the midpoint line segment is similar to the original triangle. Its three interior angles are identical to the original triangle's interior angles!
Hold on, we have one more! The four little triangles created by joining midpoints are congruent to each other! All four, including the "upside down" triangle in the middle! All congruent!
Midpoint theorem examples
Imagine or draw △RMI with midpoints E, A, and N:
Given measurements:
| Side | Length |
|---|---|
| RM | 37 inches |
| MA (full side MI) | 60 inches |
| IR | 52 inches |
Note: A is the midpoint of MI, so MA = 30 inches and the full side MI = 60 inches.
Identify the following measurements:
- RE
- EM
- MI
- AI
- IN
- NR
- EN
- EA
- NA
Did you get these answers?
| Segment | Length | Reasoning |
|---|---|---|
| RE | 18.5" | Half of RM (37 ÷ 2) |
| EM | 18.5" | Half of RM (37 ÷ 2) |
| MI | 60" | Full side (MA × 2 = 30 × 2) |
| AI | 30" | Half of MI (60 ÷ 2) |
| IN | 26" | Half of IR (52 ÷ 2) |
| NR | 26" | Half of IR (52 ÷ 2) |
| EN (midsegment ∥ MI) | 30" | Half of MI (60 ÷ 2) |
| EA (midsegment ∥ IR) | 26" | Half of IR (52 ÷ 2) |
| NA (midsegment ∥ RM) | 18.5" | Half of RM (37 ÷ 2) |
We got all that from one theorem! All that REMAINs is to summarize what you learned!
Lesson summary
| Skill | Description |
|---|---|
| Find Midpoints | Locate the unique point dividing a segment into two equal halves |
| Apply the Midpoint Theorem | Determine that the midsegment is parallel to the third side and half its length |
| Use the Converse | A line through a midpoint, parallel to a side, bisects the opposite side |
| Identify Similar Triangles | The medial triangle shares all angle measures with the original |
| Solve for Unknowns | Calculate missing side lengths or midsegment lengths using the theorem |
Now that you have carefully worked through this lesson, you are able to find the midpoint of a line segment, recall, state, and apply the Midpoint Theorem, identify similar triangles created using the Theorem, and find the unknown length of a side of a triangle (or a midpoint line segment) using the Midpoint Theorem. No magic necessary!