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Pythagorean Theorem: Why Use the Converse?

Date: 07/15/2003 at 20:46:30
From: Akima
Subject: The Converse of the Pythagorean Theorem

Dear Dr. Math -

I don't really understand the concept of the 'converse' of the 
Pythagorean Theorem. Why use the converse? What is it useful for? 

My main problem when using it is to determine if the measures given 
form a triangle, and if it is a triangle, whether or not it is a 
right tringle. For example: 5, 7, 9. I don't understand what my book 
is asking.

Another concept: The measures given form a triangle. Classify each 
tringle as right, acute, or obtuse. For example: 8, 8, 9.

- Akima

Date: 07/15/2003 at 23:29:07
From: Doctor Peterson
Subject: Re: The Converse of the Pythagorean Theorem

Hi, Akima.

You can start with our FAQ on the Pythagorean theorem itself:

   Pythagorean theorem 

The theorem says that IF a triangle is a right triangle, THEN the sum 
of the squares of the two shorter sides equals the square of the 
longest side (the hypotenuse).

The converse says the opposite: IF the sum of the squares of the two 
shorter sides equals the square of the longest side, THEN the triangle 
is a right triangle (and the angle opposite the longest side is the 
right angle).

So let's consider a triangle with sides 5, 6, and 7 (to use one 
different from your problems). First we want to know if it IS a 
triangle in the first place - can you use these lengths to make one? 
For that, you use the Triangle Inequality, which says that the sum of 
any two sides has to be greater than the third side. Imagine having 
three sticks of these lengths. Connect the two shorter sticks with a 
flexible hinge, and bend them at different angles. The distance 
between the far ends, where the third stick has to go, can vary from 
the difference of the two lengths, 6-5=1:

  +-----------+   <-- third side goes here, length 1

to the sum of the lengths, 5 + 6 = 11:

  +-----------+-------------+ <-- third side goes all the way across
        5            6

Since the longest side is less than 11, we CAN make a triangle.

Now we want to see whether the triangle is right, acute, or obtuse. 
That is, is the largest angle a right angle, or less or more than 
that? For that, we use the converse of the Pythagorean theorem. Add 
the sums of the smaller lengths:

  5^2 + 6^2 = 25 + 36 = 61

Then square the longest side:

  7^2 = 49

These are NOT equal, so it is not a right triangle.

If it HAD been a right triangle, it would be

  |  \
  |     \ sqrt(61)
 5|       \
  |         \
  |            \

Another theorem tells us that because the longest side is SMALLER than 
it has to be for a right triangle, the angle opposite it is SMALLER 
than a right angle. So this is an acute triangle; even its largest 
angle is acute, so all of them are. If the square of the longest side 
had been larger than the sum of the squares of the others, then it 
would be "too big" for a right angle, and the angle opposite it would 
be obtuse.

Here is another place to read more about this:

   Converse of the Pythagorean Theorem 

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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