Converse of the Pythagorean TheoremDate: 02/14/2003 at 00:30:29 From: Michael Subject: The Pythagorean Theorem What is the converse of the Pythagorean theorem? Date: 02/14/2003 at 08:38:48 From: Doctor Rick Subject: Re: The Pythagorean Theorem Hi, Michael. This will help you understand what a converse is: Converse, Inverse, Contrapositive http://mathforum.org/library/drmath/view/55349.html If we have a theorem that says "If A is true, then B is true," the converse of this theorem would be "If B is true, then A is true." This is a very different statement, and you can't just take any theorem and say that its converse is also true. You need to prove it as a separate theorem. The Pythagorean theorem is: In a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In other words: IF triangle ABC is a right triangle (with the right triangle at B), then AB^2 + BC^2 = AC^2. The converse of this is: If the sides of triangle ABC are such that AB^2 + BC^2 = AC^2, then the triangle is a right triangle (with right angle at B). The converse of the Pythagorean theorem allows you to determine whether a triangle is a right triangle if you know the lengths of its sides. The Pythagorean theorem itself cannot be used in this way, because you have to know that the triangle is a right triangle before you can apply it. Sometimes people miss this distinction, and they can get away with it because both theorems happen to be true. But it's important to see the difference because not every theorem is like this: it may be that a theorem is true but its converse is not. Is this what you wanted to know? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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