Common Core Standards: Math

Geometry 8.G.B.6

6. Explain a proof of the Pythagorean Theorem and its converse.

We're about to start working with the Pythagorean Theorem, which most students are somewhat familiar with, probably. (Remember it? It says, basically, that in any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.)

And, while it's called a theorem, that doesn't mean we're unsure about it. It's a fact. The Pythagorean Theorem works every single time. In fact, so does its converse, which means that we can switch the rule: if the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, then it's a right triangle. (Most of the time, the converse of a theorem isn't true—but it's true this time!)

And you believe us, right? Well, you shouldn't! Hasn't geometry taught you anything? Don't believe anything until you prove it.

Students should understand the Pythagorean Theorem and explain simple proofs of it. A good one to start out with might be drawing a 3-4-5 triangle along with squares on the side lengths, like this.

Then, using a = 3, b = 4, and c = 5, does the Pythagorean Theorem hold up? In other words, is it true that a2 + b2 = c2? Rather than squaring the numbers, students can count up the number of boxes in each side's square and see that 9 + 16 = 25. Having a protractor to verify that the triangle does have a right angle might also be helpful.

Asking students to explain the importance of certain figures and diagrams is very helpful (since that's sort of what geometry is, anyway), and drawing on prior knowledge like the distance formula, perpendicular lines, and theorems about angles is always encouraged. Given the right information, they should be able to use the Pythagorean theorem to conclude whether a triangle is right or not.

So Pythagoras was right. His theorem works. (He wouldn't be so famous if it didn't work, now would he?)

Drills

1. What does the Pythagorean Theorem tell us?

If a triangle is a right triangle with legs of length a and b and a hypotenuse of length c, then a2 + b2 = c2

You've probably heard about the Pythagorean Theorem since the third grade, but it's about time you knew exactly it means. For the Pythagorean theorem to apply, we should be talking about right triangles, so we can eliminate (B) and (C) right off the bat. While (D) is true, it applies to all triangles, not just right triangles. Only (A) is what the Pythagorean Theorem's all about.

2. What is the Converse of the Pythagorean Theorem?

If a triangle has side lengths a, b, and c such that a2 + b2 = c2, then the triangle is a right triangle

The Pythagorean Theorem says that right triangles follow the equation a2 + b2 = c2, so the converse would be about triangles following a2 + b2 = c2 being right. It says nothing about triangles not being right or not following a2 + b2 = c2, so (A) and (B) are wrong. While (D) is technically true, it's not the Converse of the Pythagorean Theorem because it assumes we have a right triangle to begin with. That leaves (C), the right answer.

3. Would the Pythagorean Theorem work with any three numbers?

No, only for sides of a right triangle

Pythagorean Theorem is all about a2 + b2 = c2, and it doesn't work unless a, b, and c are the side lengths of a right triangle. So yeah, 32 + 42 = 52 works, but so does 52 + 122 = 132, which aren't consecutive integers. To check if this works for any three numbers, we can take 8, 9, and 10. Since we can see that 82 + 92 ≠ 102, we know that (D) is the right answer.

4. Which of the following are possible side lengths for a right triangle?

6, 8, 10

Any side lengths that are possible for a right triangle must fit into the formula a2 + b2 = c2. That's what the Pythagorean Theorem tells us, anyway. In the formula, a and b should be the smaller sides and c is always the hypotenuse (the longest side). If we plug in these side lengths, we'll see that the only possibility is (C) because 36 + 64 = 100.

5. We've found that the equation 82 + 152 = 172 is true. What does this mean?

If the sides of a right triangle are of length 8 and 15, its hypotenuse is 17

How plausible are these answer options? Thanks to the right triangle with side lengths 6, 8, and 10, we know that the hypotenuse doesn't always have to be a prime number. It does, however, need to be the largest of the three side lengths, so (B) and (C) are out. While (D) is true, it has nothing to do with the 8, 15, and 17 given. Those are possible side lengths for a right triangle, with 17 as the hypotenuse.

6. Is it possible for the Pythagorean Theorem not to work for some right triangles?

No, because we have proven it for all right triangles

Let's get this straight. Theorems aren't silly ideas about mathematical concepts. Theorems are proven facts. That means the Pythagorean Theorem works for all right triangles in the whole entire universe. Proofs are the truth, so they're more everlasting than life or love (and definitely longer than most celebrity marriages).

7. Why is the Converse of the Pythagorean Theorem equally as important as the Pythagorean Theorem itself?

It allows us to identify right triangles

The Pythagorean Theorem is a conditional statement that takes the form of "right triangle → a2 + b2 = c2 satisfied." The converse switches the positions of these two arguments, so we can conclude that certain triangles are right if they satisfy the equation a2 + b2 = c2. Since "right triangle" is in the conclusion for the converse, we know that (B) is incorrect. The Pythagorean Theorem doesn't involve angle measures and its converse is important, so (A) is the only logical answer.

8. A classmate claims that he has disproved the Pythagorean Theorem. He has a right triangle whose sides are 720, 1961, and 2089. What do you think?

Get a calculator and check it again. It actually does satisfy the Pythagorean Theorem.

Honestly, did you really think anyone could disprove a theorem? Nope, not in Euclidean geometry. A quick calculator check will tell us that those three numbers work with the formula a2 + b2 = c2 (7202 + 19612 = 4,363,931 = 20892). And for the record, Hollywood probably wouldn't care either way.

9. Which of the following three numbers are also the sides of a right triangle?

If we just apply the Pythagorean Theorem to each, we'll see that only (C) works because . The others don't follow the a2 + b2 = c2 formula, and according to the Pythagorean Theorem, all right triangles must. Since they don't, we know they aren't right triangles. Simple as that.

10. Which of the following is not a way to prove the Pythagorean Theorem?