Pythagorean Theorem

Wednesday, February 18, 2009

Mr. Harbeck's last assignment posed some questions:
-Who's the guy in the picture?
-What's the formula for?
-What's with the square and the triangle?

-What does R.A.T mean?

Well, soon those questions will be answered to the best of my ability. Welcome to Geometry and the Pythagorean theorem.

To start off, the guy in the picture is Pythagoras. He was a Greek scholar who did exceptionally well in the fields of math and philosophy. He had also proved the Pythagorean theorem posed by other Pythagoreans.

The theorem said that the hypotenuse (c) is equal to the square of the sum of the other two sides (a+b2) or legs. The formula is as follows:

A2 + B2 = C2

To prove this theorem, Pythagoras took a Right Angled Triangle (R.A.T.) and squared the numbers. He then added the numbers up and got C2. To solve for C, you just square root.
If you don't get it, here's a video to help you understand.

Now here's a problem: Draw a triangle with its A side labeled 17cm ( I can't paste a picture in here, it just doesn't work for me). Then, draw a square on its hypotenuse, side c. The area of the square is 914cm2. Now, try to find the perimeter of the triangle.

Here's how you do it:

First you have to figure out the length of one side of the square. You do this by square rooting 9142. You get 914. This means the area of the square is 914x914 or 835396. The square root of that is 914 making the side of the hypotenuse 914cm.

Now you solve for B.

A2 + B2 = C2

172 + B2 =9142

B2 = 9142 - 172

B2 = 835396- 289

B2 = 835107

When you've figured out B2, square root.

Square root of B2 = B

Square root of 835107 is 913.841.

There you go! Side B on the triangle is 913.841.

Now, you're not done yet. You have to figure out the perimeter of the triangle.

We have:



C= 914cm

Add it all up and you get the perimeter: 1844.841cm

Here's another problem: A checkerboard has 64 squares in 8x8 rows. Each square is 3x3 cm long.
The questions are:
A) What is the length of the diagonal of a small square?
B) What is the length of the diagonal of the entire board?

First A. To find the diagonal of a small 3x3 square is somewhat complicated. Think of a triangle with each leg 3x3. Then, find the hypotenuse:

A2 + b2 = c2

32 + 32 = C2

32 = 9

9 + 9 = C2

C2 = 18

To solve for C, Just square root:

C = 4.242

Now we do the same but, for the whole board.

The dimensions of the board are 64 squares 3x3 each, BUT, we only need to use the square along the perimeter.

Those equal 24cm


So a triangle has legs 24x24.


242 + 242 = C2

576 + 576 = C2

C2 = 1152

Square roots mean

C = 33.941


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