Pythagorean Theorem

• This project dives into the Pythagorean theorem and its uses. It was developed by Aldo Prado in the Winter of 2015.

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a² + b² = c²

Aldo Prado

Winter 2015

Pythagoras

Pythagoras was a Greek philosopher and mathematician. He made a number of contributions to philosophy and religion, but is best known for his contributions to mathematics, specifically the Pythagorean theorem.

There is a formula that you can use in order to find the missing side of a right triangle. It is called the Pythagorean theorem.

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I know the measures of two sides of this right triangle, but not the third. I wish there was some formula to help me find the missing side.

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Proof

Construct a square with side lengths a+b (as shown here) and then connect each partition of the sides to create another square inside the bigger square and call these sides c. The area of the entire square is a²+2ab+b² (length times width). The sum of the area of each portion also describes the area of the entire square. Each triangle with sides a and b has an area of ½ ab and the smaller square has an area of c². Since there are 4 triangles, their area is 4(½ ab)=2ab. Thus, a²+2ab+b² = 2ab+c² and if we subtract 2ab from both sides, then we can conclude that a² + b² = c².

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You have to be “right”

The Pythagorean theorem only works for right triangles, that is, triangles with a 90 degree angle.

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Objectives

You will learn:

• How to determine if the triangle is a right triangle or not.

• How to label each side of a right triangle according to the Pythagorean Theorem.

• How to solve for a missing side of a right triangle using the Pythagorean Theorem.

Since the Pythagorean Theorem only works with right triangles, we need to be able to identify right triangles. The 3 triangles on top all have a right angle, denoted by the square and pointed to by the arrows. The bottom three, on the other hand, do not have a right angle, which means the Pythagorean Theorem would not work for those particular triangles.

The side opposite of the right angle in a right triangle is called the hypotenuse. We always label this side c. The two remaining sides, we label a and b; it does not matter which is a and which is b. Once we have labeled the sides, we insert the values into the equation a² + b² = c².

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Hypotenusea² + b² = c²

Let’s try it out!

First, we need to label each side. In this case a=3, b=4, and c is unknown. So, we will use Pythagorean Theorem substitute each known value into the equation a² + b² = c². Thus, 3² + 4² = c². Since 3²=9 and 4²=16, then we know 9+16= c². Next, since 9+16=25, then 25= c². The last step is to take the square root of both sides. The square root of 25 is 5 and the square root of c² is c, so 5=c. And we have found the missing side!

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?a² + b² = c²3² + 4² = c²9 + 16 = c² 25 = c² √25 = √c² 5 = c

Your Turn!

Set up the equation using the Pythagorean Theorem and solve for the missing side. If you found c to be 10, you are correct!

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Find the missing side length.

Let’s try another one!

First, we need to label each side. In this case a=5, b is unknown, and c=13. So, we will use Pythagorean Theorem substitute each known value into the equation a² + b² = c². Thus, 5² + b² = 13². Since 5²=25 and 13²=169, then we know 25+b²=169. Next, we subtract 25 from both sides and we are left with b²=144. The last step is to take the square root of both sides. The square root of b² is b and the square root of 144 is 12, so b=12. And we have found the missing side!

a² + b² = c²5² + b² = 13²25+ b² = 169-25 -25 b² = 144 √ b² = √144 b = 12

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Your Turn!

Find the missing side length.

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Set up the equation using the Pythagorean Theorem and solve for the missing side. If you found a to be 10, you are correct!

Review

The side opposite of the right angle is the hypotenuse, so we label it c. The two remaining sides, we label a and b. Once we have labeled the sides, we insert the values into the equation a² + b² = c² and solve for the missing variable by combining like terms, isolating the variable, and taking the square root of both sides.

a = 9

b = 12

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Hypotenuse

a² + b² = c²9² + 12² = c²81+144 = c² 225 = c² √225 = √c² 15 = c