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Converse of the Pythagorean Theorem

Date: 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 

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 
Associated Topics:
High School Logic
High School Triangles and Other Polygons

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