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### The Pythagorean Theorem

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Date: 07/07/97 at 13:57:09
From: Noelle
Subject: Geometry

I'm having problems with the Pythagorean Theorem! My teacher has
explained it but I still don't get it. Some of it is a breeze but
other parts just stump me!

Thanks,
Noelle
```

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Date: 09/19/97 at 13:15:00
From: Doctor Chita
Subject: Re: Geometry

Hi Noelle:

More than 2,000 years ago Pythagoras came up with an extraordinary
discovery about the relation among the squares on the sides of a right
triangle. You can understand what he discovered by following the next
four steps:

1. Draw a right triangle. The two sides that form the right angle are
called "legs". The third side, opposite the right angle, is called
the hypotenuse.

2. Label the vertices of the triangle A, B, and C, with C at the right
angle. Then label the sides opposite each vertex a, b, and c.
If you did this correctly, the two legs are a and b, and the
hypotenuse is c.

3. Now draw a square on each side of the triangle. The areas of the
squares are a^2, b^2, and c^2. (The symbol ^ means "raise to a
power." In this case, square the numbers.)

4. What Pythagoras demonstrated was that the sum of the areas of
the squares on the legs is equal to the area of the largest square.
Using our labels, you can write: a^2 + b^2 = c^2. This is
Pythagoras' theorem.

It's important to remember that we don't have to use A, B, and C (and
a, b, and c) as our labels. Any labels will do. In general the Theorem
says that:leg(1)^2 + leg(2)^2 = hypotenuse^2. When using the theorem
to solve problems it's important to locate the right angle in the
triangle and then identify the legs and the hypotenuse. Once you have
those, you can use the formula.

You can solve many problems involving right triangles by knowing any
two of the three sides. For example, suppose the sides you are given
are the two legs, and they measure 3 and 4.  Then the unknown side is
the hypotenuse.

Substitute 3 and 4 for a and b in the theorem and solve for c.

3^2 + 4^2 = c^2
9 + 16 = c^2
25 = c^2
5 = c

(Actually, there are two answers to c^2 = 25. One is 5 and the other
is -5. But in geometry we only use positive lengths so we can
disregard -5.)

If the two sides you are given are a leg and the hypotenuse, then to
find the length of the unknown leg, you have an extra algebra step to
complete.

Let's say a = 5 and c = 13, where a is a leg and c is the hypotenuse.
Then,

a^2 + b^2 = c^2
5^2 + b^2 = 13^2
25 + b^2 = 169

Subtract 25 from both sides of the equation:

b^2 = 169 - 25 = 144
b = 12. (Throwing away -12 as a length.)

Does this help?

-Doctor Chita,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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