**THE PYTHAGOREAN THEOREM**

**Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

**Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!**

**1. Memorize and apply the Square Root Property
2. Memorize and apply the Pythagorean Theorem**

**The Square Root Property**

The **Square Root Property** is most often used when solving certain quadratic equations. Specifically, those of the form **a x^{2} = c** where

The **Square Root Property** states the following:

If

is an algebraic expression containing a variable andudis a constant, then has exactly two solutions, namely and or simply .

**Example 1:**

Solve the quadratic equation .

First, we isolate the squared term to get

Now, we use the **Square Root Property**. In this case, the * x* in the equation is the

That is,

then and

or

Note that when evaluated with a calculator. There are infinitely many decimal places which means that we are dealing with an irrational number.

**Example 2:**

Solve the quadratic equation .

First, we isolate the squared term to get .

Then we use the **Square Root Property**.

That is,

then and

or

Please note that we have an integer solution.

**The Pythagorean Theorem**

The right triangle is associated with one of the most
famous and useful theorems (assertions that can be proved true using the
rules of logic) in mathematics. It is called the **Pythagorean
Theorem** named for the Greek mathematician Pythagoras.

The *Pythagorean
Theorem* is probably one of the most important mathematical equation used
in the building trade. Due
to is, drawings can be enlarged, foundations can be build, and
square angles can be calculated. It
can be used by everyone from surveyors who want to find out how tall
a mountain is to astronomers who want to calculate the distance to
the sun or the circumferences of the moon, to weathermen who try to
calculate the height of clouds. Carpenters will use it to keep their work square and
draftsmen use it to ensure the accuracy of their architectural
drawings.

The *Pythagorean
Theorem* states that the square of the hypotenuse of a right
triangle is equal to the sum of the squares of the two legs of the triangle.

The **Pythagorean Theorem** is usually
presented as follows:

, where * a*and

**
Example 3:**

Given a right triangle, assume that the lengths of the
legs are * a = 6*
and

Using the *Pythagorean Theorem
*,
we get

We are now dealing with a quadratic equation where * c* is the variable. We use the

NOTE that we are not using the negative solution as shown in Examples 1 and 2 since the sides of triangles are never negative.

Using the calculator, we find There are infinitely many decimal places which means that we are dealing with an irrational number.

**
Example 4:**

Given a right triangle, assume that the length of the
leg *b*** **is

Using the *Pythagorean Theorem
*,
we get

We are now dealing with a quadratic equation where ** a** is the variable. We isolated the variable to get

Finally, using the *Square Root Property* we get

NOTE that we are not using the negative solution as shown in Example 1 and 2 since the sides of triangles are never negative.