 Hello Geometers, today we're going to talk about the converse of the Pythagorean theorem. As you all know, the normal Pythagorean theorem says a squared plus b squared equals c squared. And this is true in a right triangle. But today we're going to talk about the converse of the Pythagorean theorem. Converse means switch. I don't know if you remember this from when we learned logic. But converse means switch. So although it may not seem like a big deal, we're going to switch the formula to say that c squared equals a squared plus b squared. And if those two sides are equal, then we still have a right triangle. If those two are not equal, then we have either an obtuse triangle or an acute triangle. So we're going to do some sample problems to find out how we can tell whether a triangle is right, acute, or obtuse by using this converse of the Pythagorean theorem. So let's say I have a triangle that has sides 16, 12, and 20. The first thing I'm going to want to do is make sure it really is a triangle. So I take my 16 and my 12, my two shorter sides, I add them together, I get 28. And 28 is indeed larger than 20. So yes, it makes a real triangle. Now I set up my equation, I take my longest side, that's the c of the formula, equals the a squared plus the b squared. You pause the video now and do the numbers, do the math, and see what kind of triangle you think you have. Pause the video and do the arithmetic here. I shouldn't have used the equal sign here between the two sides of the equation because I don't know if they're equal, I should have used a question mark here. In this case, once I do all my arithmetic, I end up with the fact that 400 equals 400. So in this example, the triangle is a right triangle. Now I'm going to try the same process with another triangle. In this triangle, the sides are 4, 2, and 3. I'm going to put them into my formula to see what kind of triangle I have. No, first I'm going to see if they are a triangle. Here my two shorter sides are 2 and 3. 2 plus 3 is 5 and 5 is greater than 4. So yes indeed, it is a triangle. So I have my formula here. I do not know what kind of symbol will go between them, but I have my longest side being my C squared and then I have A squared plus B squared. So now pause the video and do the math and see how those two sides of the expression compare. In this case, once I've simplified both sides of the formula, I find that 16 is greater than 13 because my larger side is greater than if it were a right triangle. That means that this is an obtuse triangle. Now the last example I'm going to try is going to have sides 5 and 6 and 4 long. Again, first thing I have to do is make sure it's a triangle. 5 and 4 are the short sides. 5 plus 4 is 9. 9 is greater than 6. So yes, I do have a triangle. So filling in my equation, I take the longest side, the 6 and then the A squared plus B squared and I'm going to see how those compare. At this point, you pause the video and work out the arithmetic yourself and see what you get for an answer. Pause the video. So when I do this work, I find that 36, the C squared is less than the right side of the equation, what it would be if it were a right triangle. So because 36 is less than what I thought it was going to be, then I expected or less than the right triangle. This triangle is an acute triangle. So those are some examples of how you can use the converse of the Pythagorean Theorem to decide whether a triangle is an acute triangle, a right triangle, or an obtuse triangle.