Lesson 11-4: The Midpoint Formula (Day 114)

Lesson 11-4 of the U of Chicago text is called "The Midpoint Formula." In the modern Third Edition of the text, the midpoint formula appears in Lesson 11-7.

Like Lesson 11-1, Lesson 11-4 wasn't truly covered on the blog two years ago. I actually did a better job covering Chapter 11 three years ago than two years ago.

And so this is what I wrote three years ago about today's lesson:

Lesson 11-4 of the U of Chicago text covers the other important formula of coordinate geometry -- the Midpoint Formula. As the text states, this is one of the more difficult theorems to prove.

In fact, the way we prove the Midpoint Formula is to use the Distance Formula to prove that, if M is the proposed midpoint of PQ, then both PM and MQ are equal to half of PQ. The rest of the proof is just messy algebra to find the three distances. The U of Chicago proof uses slope to prove that Mactually lies on PQ. Since we don't cover slope until next week, instead I just use the Distance Formula again, to show that PM + MQ = PQ, so that M is between P and Q. The algebraic manipulation here is one that's not usually used -- notice that instead of taking out the four in the square root of 4x^2 to get 2x (as is done in the last exercise, the review question), but instead we take the 2 backwards inside the radical to get 4, and then distribute that 4 so that it cancels the 2 squared in the denominator.

I don't have nearly as much to say about the Midpoint Formula as the Pythagorean Theorem and its corollary, the Distance Formula. To me, it's a shame that I had to bury the Pythagorean Theorem in the middle of this Coordinate Geometry unit. The main theorem named for a mathematician really deserves its own lesson, but due to time constraints I had to combine it with the Distance Formula the way I just did it in yesterday's lesson.

Returning to 2018, this year I gave the Pythagorean Theorem and the Distance Formulas two separate lessons, but other than that I don't have anything else to say about Lesson 11-5. And I have nothing else to say (not even any excuse to bring up the IB or some other contrived topic again), and so this is a rare short post.