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Converse of the Pythagorean Theorem - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

If we know that the sum of two square sides in a triangle equals the square of the third side, we can conclude that the triangle is a right triangle. Pythagorean triples are commonly seen side lengths that indicate when a triangle is a right triangle. Knowing these triples will save time when in calculations related to the Pythagorean theorem.

The Pythagorean Theorem says that if you have a right triangle - so if I draw this in, a right triangle with legs A and B and hypotenuse C then if it's a right triangle A squared plus B squared must equal C squared. But what about the converse of the Pythagorean Theorem? Remember a converse switches the if and then parts of a conjecture. So what if we didn't know that this was a right angle? What if I told you that A squared plus B squared equals C squared? So that's what the converse is saying, that if you have a triangle with the sides A, B, and C if A squared plus B squared equals C squared then you can assume something. And that is that we can assume it's a right triangle.
Then it is a right triangle.
So if A squared plus B squared equals C squared then any triangle you have a right triangle. That's the converse of the Pythagorean Theorem.
So concrete examples of the converse are the Pythagorean triples. There are four that you should memorize, "3, 4, 5", "5, 12, 13", "8, 15, 17" and "7, 24, 25". So what these triples are is an application of the converse, which means three squared plus four squared equals five squared and nine plus 16 equals 25. So that's true for all four sets of these triples, that when you apply the Pythagorean Theorem it works.
Now, you don't have to just memorize these but you have to be realizing that if you multiply all of these by two so you get a 6, 8, 10 that all the multiples of these triples also apply. So let's say you multiply these by 3, you have 9, 12, 15. Or if you multiply them by four you would have 12, 16, 20.
So by memorizing the original triple you can just multiply and recognize the same triple. So that works for all of these. If you multiplied that triple by two you would have 10, 24, 26.
So the key thing about memorizing these is if you are given two and asked to find the third, and you recognize it as a triple, so let's say your given a triangle with sides 10 and 24 and your told that it's a right triangle, you don't even have to do the Pythagorean Theorem. You can just say well this is a Pythagorean Triple. That's 10, 24, 26.
So memorizing these will save you a lot of time when you're trying to find missing sides and missing hypotenuse.

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