## Introduction: Pythagorean Theorem

The name Pythagorean theorem came from a Greek mathematician by the named Pythagoras. Pythagoras developed a formula to find the lengths of the sides of any **right triangle**. Pythagoras Discovered that if he treated each side of a right triangle as a square (see figure 1) the two smallest squares areas when added together equal the area of the larger square. The formula is A2 + B2 = C2, this is as simple as one **leg **of a triangle squared plus another leg of a triangle squared equals the **hypotenuse **squared.

In this lesson I will teach you how to use the Pythagorean theorem, I will show where you put it to use and some different ways to use the theorem to find the lengths of legs when given the leg length and the hypotenuse length. I will try my best to explain every step of the way to my fullest and complete answer.

My inspiration for this instructable came from having the interest of finding how formulas work. I take interest especially in the Pythagoras theorem because we use it in lots of every day jobs such as engineering, woodworking and metalworking. I hope I can pass my interests on to you in this lesson.

Prescribed learning outcomes, by learning how the Pythagorean theorem works students will learn how to square, square root, add and subtract, and learn the Pythagoras formula.

Key words...**Hypotenuse- **In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle.**Leg**- Either sides of a right triangle that are opposite to the hypotenuse.**Right triangle**- A triangle that has one corner of a ninety degree angle.

## Step 1: How to Use the Formula

Lets start with an example. If we know that leg A of the triangle is 3cm and leg B is 4cm the first step is **squaring** our legs. We can do this simply by multiplying one leg by same amount as its self, So therefore we get A = 9 and B = 16. The next step is adding we have to add both squared legs together to get one number witch in our case is 25. the final step is finding your **square root** of this final added number in this case is 5. Now that we have done every step we can come to the conclusion that the hypotenuse is 5cm.**Review,** in doing this equation is is extremely important that we follow all steps exactly. When learning the formula you have to have a basic understanding of three things, how to square, how to square root and how to tell which side is the hypotenuse. A little trick I use to find the hypotenuse is the two little lines that symbolize the ninety degree angle point to the hypotenuse.

Key words...**Squaring-** Squaring is the number you get when you multiply the number by it by its self.**square root- **A divisor of a quantity that when squared gives the quantity. For example, the square root of 25 is 5.

## Step 2: The Second Method

There are two methods in the theorem one you are given both the lengths of the legs and the other you are given the length of one leg and the hypotenuse. So for example our C side equals 12 and our B side equals 6. So if we know that C2 = 144 that means that A2 + B2 = 144 therefore 6 squared (36) plus B squared = 144. Now the next step is finding B2 the way you do that is to take both your known leg and the hypotenuse and subtract in our case we go B = 144 - 36 therefore B = 108.**review- **In this method we have to remember that we are trying to find a leg not the hypotenuse. We have to remember that instead of using the formula the way its written we have to order the steps differently, this order is C2 - B2 = A2.

Key words...**Hypotenuse**- The side opposite to the legs of the triangle( hint its the longest side)

## Step 3: Pythagorean Triples

A **Pythagorean triple **is any group of three** integer** values that satisfies the equation a2 + B2 = C2 is called a Pythagorean triple. therefore any triangle that has sides that form a Pythagorean triple must be a right triangle. When all three sides are **whole numbers** you have a Pythagorean triple*.* For example A = 3 B = 4 C = 5 this can also be called a 3,4,5 triangle. Here is how you do the equation for example 3 squared plus 4 squared = 5 squared, in other words 9 + 16 = 25 therefor because these are all whole numbers the triangle must be a Pythagorean triple.

There are four main **Pythagorean triples families** there is the 3,4,5, the 6,8,10, the 5,12,13, and the 8,15,17 triangles. If you multiply any of the three integers by the same amount you will still have a Pythagorean triple. For example 3,4,5, multiplied by two will give you 6,8,10, witch is a Pythagorean triple.**Review- **The integers represent the lengths of the sides of the triangles in a,b,c, order. If you do the equation and you don't come out with a whole number the integers are not a Pythagorean triple. Remember when multiplying Pythagorean triples families multiply all three numbers by the same amount.

Key words...**Pythagorean triple-**A right triangle where the sides are in the ratio of integers. (Integers are whole numbers like 3, 12 etc)**integer-**Includes the counting numbers {1, 2, 3, ...}, zero {0}, and the negative of the counting numbers {-1, -2, -3, ...}**Whole numbers-**There is no fractional or decimal part. And no negatives.

Example: 5, 49 and 980 are all whole numbers.**Pythagorean triples families-**every triple is a whole number multiple of the base triple.

## Step 4: Creat and Solve

Now using What you have just learned Cut out a A=5cm by B=12cm right triangle (don't forget to use a protractor to measure your angle) and try to figure out what side C equals. Put your answer in the comments if you like.

Now try to figure out a word problem...

If jimmy have a ladder leaning against a wall that is five feet in length, and the feet of the ladder are three feet away from the wall how far up the wall is the ladder? Feel free to answer in the comments below.**Practice questions...**

A = 6, B = ?, C =10

A = 5, B = 12, C = ?

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