Real Numbers and The Pythagorean Theorem

Real Numbers and The Pythagorean Theorem

Intro to Chapter 7

Today we will cover the following

• Square Root and Perfect Square

• Radical Sign and Radicand

• Legs and Hypotenuse

• Pythagorean Theorem

  
  

• Pythagorean Triples

Square Roots and Perfect Squares

• A square root is a number that, when multiplied by itself, equals the given number. Every positive has a positive and negative square root.

• A perfect square is a number with integers as its square root.

• Being familiar with your multiplication tables will help!

Square root

• You should ask, "do I know any numbers that can be multiplied by themselves to give me 49?

• yes, 7 x 7 = 49

• But, is there anything else that can be multiplied by itself to give us 49?

• YES! -7 x -7 = 49

Identifying parts

• The radical symbol is used to represent a square root.

• The number under the radical symbol is called the radicand

• Like the picture says, if there is no number for the index, it is implied 2.

Multiple Select

What are the square roots of 144?

• 11

• 12

• -12

• 144

Multiple Choice

In this picture, what is our radicand?

• 2

• 13

Multiple Choice

In this picure, what is our index?

• 3

• 13

• 2

Right Triangle

• Right triangles typically are indicated by the square in a corner

• Legs are two sides that form the right angle (a and b)

• Hypotenuse is the side opposite od the right angle (c)

Pythagorean Theorem

• In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Important facts about Pythagorean Theorem

• Not all sides are integers

• If you are only given a leg measurement and the hypotenuse, you can find the measurement of the missing leg by setting up an equation solving the equation.

Example Problem with missing leg

• Here we have a leg (2.1cm) and the hypotenuse (2.9cm)

•  $a^2+b^2=c^2$  

•  $a^2+2.1^2=2.9^2$  

•  $a^2+4.41=8.41$  

•  $a^2=4$  

•  $a=2$  our missing leg measurement is 2cm

Example Problem with missing Hypotenuse

•  $8^2+15^2=c^2$  

•  $64+225\ =\ c^2\ ;\ 289=c^2$  

•  $17=c$  

• Measurement of the hypotenuse is 17ft.

Fill in the Blanks

What is the measurement of the hypotenuse?

Fill in the Blanks

What is the measurement of the hypotenuse?

Pythagorean Triple

• a set of three positive integers that follow the rule of

   $a^2+b^2=c^2$  

Pythagorean Triples

• Thankfully, there is a list of Pythagorean Triples!

• This list is MASSIVE and will be given to you! But you are responsible for being able to identify items on it!