Pythagorean Theorem Ramzy Alseidi

Pythagoras is often referred to as the first pure mathematician. He was born on the island of Samos, Greece in 569 BC. Various writings place his death between 500 BC and 475 BC in Metapontum, Lucania, Italy. Pythagoras was well educated. He was interested in mathematics. We remember him today mainly for his equation relating the lengths of the legs of a right triangle to the length of its hypotenuse. The Pythagorean Theorem, also called the Pythagorean Property, says that the sum of the squares of the lengths of the legs of any right triangle is equal to the square of the length of the hypotenuse, a2 + b2 = c2. Pythagoras 90°

A right angle is an internal angle which is equal to 90°. A right angle can be in any orientation or rotation as long as the internal angle is 90°. Note the special box like symbol in the angle. If you see this, it is a right angle. 90° Angles 90°

Legs and Hypotenuse The hypotenuse is the largest side in a right triangle and is always opposite the right angle. (Only right triangles have a hypotenuse). The other two sides of the triangle are referred to as the 'legs'.

In mathematics, a square number or perfect square is a number that is the square of a number; in other words, it is the product of some number with itself. For example, 4 is a square number, since it can be written as 2 × 2 => 2 2. Perfect Squares A b C a a a b b b C x C = C 2 C C C a x a = a 2 b x b = b 2 a 2 + b 2 = c 2

The pythagorean theorem formula is a 2+ b 2= c 2. On this right angle we have the sides labeled and we will plug them accordingly to the formula. The Formula

Example Find the length of the pyhotenuse if the lengths of the legs are 5 and 12 C 2 = a 2 + b 2 (hypotenuse) = (leg) + (leg) X 2 = X 2 = X 2= +169 X=√169 X=

8 real life example the house which contains two right triangles all with the box indicating a 90º angle. the picture to show that right triangle present in our life. C 2 = a 2 + b 2 C 2 = C 2 = C 2= +25 C=√25 C= 5

Lets look at examples of triangles and decide if they are right angles or not, and if the labels given are correct for the leg’s and hypotenuse. Let’s look at some examples; Ex: a- is a right triangle and the label for the hypotenuse is correctly placed. Ex:b- is a right triangle and the label given is not correct it should be hypotenuse instead of leg. Ex:c-is not a right angle. Ex:d- is not a right triangle. Ex:e- is a right angle and the label for the leg is correct. Ex:f- is a right triangle and the label for the leg is correctly placed. Checking for Understanding

Now we know how to find which triangles are right triangles, the legs and hypotenuse, and how to plug the numbers given in to the formula so, let’s put the formula to work! Use the Pythagorean Theorem Formula

Solve for C We will work out this problem together. Solve for C: Identify the legs and the hypotenuse of the right triangle. In this case the legs have a length of 3 and 4. C is the hypotenuse because it is opposite to the right angle. The numbers to plug in to the formula will be the following: Substitute values in to the formula a 2 + b 2 = c 2 remember “c” is for the hypotenuse and will remain as c 2. Your problem should look like = c 2

Solve for C We will work out this problem together. Solve for C: Take your problem = c 2 and begin to solve for C. three square equals 9 four square equals 16 You then format your problem as 9+16= c 2 You will add 9+16 and leave it equal to c 2 This will leave you with 25= c 2 You then take the square root of both sides √25= √ c 2 and solve the square root of 25 is 5 and the square root of c 2 is C This gives you 5=c or c=5

13 Check Your Answer Plug in all the given number’s to the formula a 2 + b 2 = c 2 a= 3 b= 4 c= 5 C=5 is the correct answer. The length of the hypotenuse is 5. if a 2 + b 2 gives you the same number as c 2 then your answer for c is correct gives you 25 and 5 2 gives you 25. in this case gives you the same number as 5 2 there for your answer for c is correct.

Worksheet Remember that when you solve for a or b you must move all numbers to one side and leave c by its self to solve for the missing side.