Pythagorean Theorem

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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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19 Comments

does the Pythagorean theorem have 3 whole numbers? or the Pythagorean triple have 3 whole numbers? Can Pythagorean theorem have a non perfect whole number? or can Pythagorean triple have a non perfect whole number?

does the Pythagorean theorem have 3 whole numbers? or the Pythagorean triple have 3 whole numbers? Can Pythagorean theorem have a non perfect whole number? or can Pythagorean triple have a non perfect whole number?

You mistake in the Pythaborean triples : You say there are four basic sets; that is not true there are an infinite set.
Secondly, you mention 3,4,5 (which is correct) followed by 6,8,10 which is no basic set (because all numbers are just the same multiple of 3,4,5).
A set could be 7,24,25 (you could take it as the third set).
There is a wiki about "formulas for generating pythgorean triples". (of which i derived the set here with the 7 ;)

One of the rules (from Euclides) :
-take an uneven number (a)
-square it. You get an uneven number.
-take the two concutive numbers that sum up to the square. These are b (the smallest, even) and c (de lagest, uneven).
Since there is an infinite set of prime numbers (who are no multiple of anything) to start with as a, this proves you have an infinite set of basic pythagorean triples.
Overall a nice instructable

exactly dude (6,8,10) is not a primitive pythagorean triplet)

Sorry to point this out, but in the first picture you describe the first triangle as A=3 and B=4, but in the second picture you describe it as A^2=16 and B^2=9. So I'm unsure if the two photographs are two different triangles, or if you made a mistake, but I think it would be less confusing if the two picture matched.

Its all good i just put the picture for an example. Thanks anyway for the advise and don't forget to go vote for the cookie contest and the teacher contest

In step 2 you subtracted 36 from 144 and gave 112 as the answer. That should be 108.

Thanks so much that was a typo i'll fix it right away.

All fixed!! Do you have any siggestions to make this ible better im trying to make it the best that it can be for the teacher contest!!

Thanks, cobalt420