Mid Point Theorem

Mid Point 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 seems to work almost like an illusion, but it is both real and useful.

What is a Midpoint?

The midpoint of a line segment, such as the line segment making a side of a triangle, is the single point an equal distance from both ends of the line segment. Look at line segment $SK$ below. Between endpoints $S$ and $K$ are three other points, $N$, $A$, and $C$. Which point is the midpoint of line segment $SK$?

[insert line segment with endpoints $S$, $K$, Point $N$, then midpoint $A$, then Point C]

Point $A$ is halfway between Points $S$ and $K$. For any line segment, only one midpoint exists.

The Midpoint Theorem

The Midpoint Theorem tells us that the line segment joining two sides of any triangle at their midpoints is parallel to the third side, and the line segment is half the length of that 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.

Here is $△NET$ with side $ET = 38 meters$:

[insert scalene $△NET$ with sides NE = 25 m, $ET$ = 38 m, $TN$ = 32 m, or use a video for the next steps]

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 $G$:

[add to drawing or continue animation]

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, $AG \parallel NE$. 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 (magically?) intersect the midpoint of the remaining side.

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 angle'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

Here is $△RMI$ with midpoints $E$, $A$, and $N$:

[insert scalene $△RMI$ with Point E midway RM, Point A midway MI, Point N midway IR; label side RM = 37 inches, MI = 60 inches, IR = 52 inches]

Given measurements:

$RM = 37 inches$

$MA = 30 inches$

$IR = 52 inches$

Identify the following measurements:

1. $RE$
2. $EM$
3. $MI$
4. $AI$
5. $IN$
6. $NR$
7. $EN$
8. $EA$
9. $NA$
10. Name the three sets of parallel lines
11. Name any two similar triangles
12. Name any two congruent triangles

Did you get these answers?

1. $RE = 18.5"$
2. $EM = 18.5"$
3. $MI = 60"$
4. $AI = 30"$
5. $IN = 26"$
6. $NR = 26"$
7. $EN = 30"$
8. $EA = 26"$
9. $NA = 18.5"$
10. 1)$RM \parallel NA$; 2) $EA \parallel RI (or IR)$; 3) $EN \parallel MI$
11. $△RMI \sim △REN \sim △EMA \sim △NAI \sim △ENA$
12. $△REN \cong △EMA \cong △NAI \cong △ENA$

We got all that from one theorem! All that REMAINs is to summarize what you learned!

Lesson Summary

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!

Next Lesson:

Perpendicular Bisector