In this last section we present a few problems that require the use of the Pythagorean Theorem. Try to resolve as many problems as you can.

1) The Pythagoras Proof:

This is a hands-on exercise for you to convince yourself that the Pythagorean theorem works. It is based on the actual proof that is attributed to Pythagoras. On the following figure we have a right triangle with a square associated with each of its sides:

Using the dimensions associated with the three sides, calculate the area of each of the squares. Then make sure that the area of the hypotenuse's square (brown) equals the areas of the other two squares together.

Now for the hands on part. Draw an equivalent picture on a piece of paper. You can use any size triangle as long as it is a right triangle. Cut up and reassemble the two small squares to form a square identical to the larger one.

There is a building with a 12 ft high window. You want to use a ladder to go up to the window, and you decide to keep the ladder 5 ft away from the building to have a good slant. How long should the ladder be?

3) Baseball diamond:

On a baseball diamond the bases are 90 ft apart. What is the distance from home plate to second base in a straight line?

4) Equilateral triangle:

An equilateral triangle has vertices at (0,0) and (6,0) in a coordinate plane. What are the coordinates of the third vertex? You may want to sketch it out.
Note: The sides of an equilateral triangle are identical in length.

5) An algebraic problem:

Find out the length of sides a and b on the following triangle:

6) An iterative problem:

Look at the following figure. Start by finding the value for X₁, then for X₂, then X₃, and so on until you get the value for X₆. Write the lengths as square roots, as that makes it simpler.

What is the value of X₆?

7) A 3D problem:

We have a wooden box that measures 4 ft. by 3 ft. by 2 ft.:

Figure out: What is the longest straight pole, like the red one, that you can have inside the box?

8) Pythagorean Triples:

The Pythagorean Triples were described with Tip number 1. Here we will describe a method to generate all of the Pythagorean Triples. There is a simple formula that gives all the Pythagorean triples. If m and n are two positive integers and m < n, then the triples can be generated with the following equations:

It's easy to check algebraically that the sum of the squares of a and b is the same as the square of c.

Now try it out and produce as many triples as you can by substituting any positive integers for m and n (as long as m < n). If you know how to use a spread sheet program (like Excel), you can very quickly do a table that generates tons of triples.