Lesson 9

This warm-up is preparation for the argument of the converse of the Pythagorean Theorem that we will construct in the next activity. The warm-up relies on the Pythagorean Theorem and geometrically intuitive facts about how close or far apart the two hands of a clock can get from one another.

Arrange students in groups of 2. Give students 1 minute of quiet think time followed by partner and then whole-class discussions. 

Invite students share their solutions. 

Make the following line of reasoning explicit:

Imagine two hands starting together, where one hand stays put and the other hand rotates around the face of the clock. As it rotates, the distance between its tip and the tip of the other hand continually increases until they are pointing in opposite directions. So from one moment to the next, the tips get farther apart.

(Proving this requires mathematics beyond grade 8, so for now we will just accept the results of the thought experiment as true.)

This activity introduces students to the converse of the Pythagorean Theorem: In a triangle with side lengths \(a\), \(b\), and \(c\), if we have \(a^2+b^2=c^2\), then the triangle must be a right triangle, and \(c\) must be its hypotenuse. Since up until this unit we rarely phrase things as a formal theorem, this may be the first time students have directly considered the idea that a theorem might work one way but not the other. For example, it is not clear at first glance that there is no such thing as an obtuse triangle with side lengths 3, 4, and 5, as in the image. But since \(3^2+4^2=5^2\), the converse of the Pythagorean Theorem will say that the triangle must be a right triangle.

The argument this activity presents for this result is based on the thought experiment that as you rotate the sides of length 3 and 4 farther apart, the distance between their endpoints also grows, from a distance of 1 when they are pointing in the same direction, to a distance of 7 when pointing in opposite direction. There is then one and only one angle along the way where the distance between them is 5, and by the Pythagorean Theorem, this happens when the angle between them is a right angle. This argument generalizes to an arbitrary right triangle, proving that the one and only angle that gives \(a^2+b^2=c^2\) is precisely the right angle.

The next activity will more heavily play up the distinction between the Pythagorean Theorem and its converse, but it is worth emphasizing here as well.

Arrange students in groups of 2. Give students 2 minutes of quiet work time. Ask partners to share their work and come to an agreement. Follow with a whole-class discussion.

Select some groups to share the reasoning. It is important to get out the following sequence of ideas:

• Just as we saw with the clock problem, the length of the third side continually increases from 1 to 7 as the angles increases between \(0^\circ\) and \(180^\circ\).
• Because of this, there is one and only one angle along the way that gives a third side length of 5.
• By the Pythagorean Theorem, if the angle is a right angle, the third side length is 5.

Combining these, we see that the one and only angle that gives a third side length of 5 is the right angle. That is, every triangle with side lengths 3, 4, and 5 is a right triangle with hypotenuse 5. Triangles like the one displayed here are thus impossible.

As a class, discuss how the argument could have been run with any two starting side lengths \(a\) and \(b\) instead of specifically \(3\) and \(4\).

Though we do not need to write it as formally, the length of the third side continually increases from \(a-b\) (or \(b-a\)) up to \(a+b\) as the angle increases between \(0^\circ\) and \(180^\circ\). When the angle is a right angle, the third side has length equal to the value of \(c\) that makes \(a^2+b^2=c^2\), and as in the previous discussion, that is the only angle that gives this length.

We conclude that the only way we could have \(a^2+b^2=c^2\) is if the triangle is a right triangle, with hypotenuse \(c\). This result is called the converse of the Pythagorean Theorem. Together, the Pythagorean Theorem and its converse provide an amazing relationship between algebra and geometry: The algebraic statement \(a^2+b^2=c^2\) is completely equivalent to the geometric statement that the triangle with side lengths \(a\), \(b\), and \(c\) is a right triangle.

The purpose of this activity is for students to apply the Pythagorean Theorem and its converse in mathematical contexts. In the first problem, students apply the Pythagorean Theorem to find unknown side lengths. In the second problem, students use the fact that by changing side lengths so that they satisfy \(a^2+b^2=c^2\), the resulting triangle is guaranteed to be a right triangle.

Provide students with access to calculators.

The goal of this discussion is for students to use the Pythagorean Theorem to reason about the side lengths of right triangles. Select 2–3 students to share their work for the two problems.

For the second problem, make a list of all possible changes students figured out that would make the triangle a right triangle by changing the length of just one side. Depending on where students decided the right angle is located and which side they selected to change, there are nine possible solutions. Can students find them all?