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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

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.

Pythagorean theorem
http://mathforum.org/dr.math/faq/faq.pythagorean.html

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:

5
+-----------+   <-- third side goes here, length 1
+--------------+
6

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|       \
|         \
|            \
+--------------+
6

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.

Converse of the Pythagorean Theorem
http://mathforum.org/library/drmath/view/62215.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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