@rebrokeraaron#duet with @dustin_wheeler

I came across a nice TikTok vid recently that definitely screamed out for a mathematical explanation. It was interesting to watch because I had written an activity not too long ago which connected the two mathematical ideas relevant to this video—I just didn’t have this nice context to apply it to.

Anyway, you can see it at the right there.

The two connected ideas at work here are the Triangle Proportionality Theorem and the Midpoint Formula. We don’t actually need both of these concepts to explain the video. It’s just more pleasing to apply them both together.

The Triangle Proportionality Theorem (see an interactive proof here) tells us that if we draw a line segment inside a triangle that connects two sides and is parallel to the third side, then the segment will divide the two connected sides proportionally.

So, \(\mathtt{\overline{DE}}\) above connects sides \(\mathtt{\overline{AB}}\) and \(\mathtt{\overline{AC}}\), is parallel to side \(\mathtt{\overline{BC}}\), and thus divides both \(\mathtt{\overline{AB}}\) and \(\mathtt{\overline{AC}}\) proportionally, such that \(\mathtt{a:b=c:d}\).

And of course the Midpoint Formula, which we’ll suss out in a second, is \(\mathtt{\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)}\).

Okay, Let’s Explain This

What’s happening in the video is that the measurer is creating a right triangle. They first make a horizontal measurement—which they don’t know the midpoint of. That’s the horizontal leg of the right triangle. Then they take the tape measure up the board diagonally. That’s the hypotenuse of the right triangle. We **do** know that midpoint (because it’s a simpler whole number).

We can drop a line segment from the midpoint that we know straight down vertically, such that the segment is parallel to the vertical side of the triangle (the vertical side of the board).

Now we have set up everything we need for the Triangle Proportionality Theorem. The vertical segment connects two sides—the hypotenuse and the horizontal side—and is parallel to the third (vertical) side. Thus, the vertical segment divides the hypotenuse and horizontal segment into equal ratios.

But since we know that the segment is at the midpoint of the hypotenuse, we know that it divides the hypotenuse into the ratio \(\mathtt{1:1}\). That is, it divides the hypotenuse into two congruent segments. Therefore, it must also divide the horizontal line segment into two congruent segments!

Okay, so that pretty much nails it (ha!). Why do we need the Midpoint Formula?

Now for the Midpoint Formula

Well, we don’t really need the Midpoint Formula. It’s just interesting that the Triangle Proportionality Theorem explains why the Midpoint Formula works. Working through the explanation above, it may have occurred to you that what the Triangle Proportionality Theorem says, indirectly, about ANY right triangle, is that—if you can wrangle the legs to be horizontal and vertical—the midpoint of the hypotenuse is always directly above or below the midpoint of the horizontal side, and the midpoint of the hypotenuse is always directly to the left or right of the midpoint of the vertical side. That’s what the Midpoint Formula basically says in symbols.