# 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 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.

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** 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.

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 (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

Imagine or draw **△RMI** with **midpoints E**, **A**, and **N**:

Given measurements:

**RM = 37 inches****MA = 30 inches****IR = 52 inches**

Identify the following measurements:

**RE****EM****MI****AI****IN****NR****EN****EA****NA**

Did you get these answers?

**RE = 18.5"****EM = 18.5"****MI = 60"****AI = 30"****IN = 26"****NR = 26"****EN = 30"****EA = 26"****NA = 18.5"**

We got *all that* from one theorem! All that **REMAIN**s 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!