The Pythagorean Theorem


 

Unit Objective

●    Find Principal (positive) square roots and find missing sides of a right triangle using the Pythagorean theorem.

 

Key Vocabulary

●    Square root - The square root of a number is a value that, when multiplied by itself, gives the original number.
In other words, the square root of a number "a" is a number "b" such that b multiplied by itself equals a. The symbol used to represent the square root of a number is √, and the expression "√a" is read as "the square root of a." For example, the square root of 9 is 3, because 3 multiplied by itself equals 9: √9 = 3

 

Right triangle - A triangle in which 1 of the angles is a 90 degree angle (right angle).


 

 

Let's Practice.
Which triangles are right triangles?

 

 

 

 

 

Perfect Squares

 

 

 

Perfect squares are integers that are the result of squaring another integer. In other words, a perfect square is a number that can be expressed as the square of some other integer. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on are perfect squares, because each of them is the result of squaring some integer.

The sequence of perfect squares goes on infinitely, and it can be represented by the formula n^2, where n is a positive integer. For instance, the first few perfect squares can be found by calculating 1^2=1, 2^2=4, 3^2=9, 4^2=16, and so on.

Perfect squares are used in many areas of mathematics, including geometry, algebra, and number theory. They also have practical applications in fields such as physics, engineering, and computer science.

 

 

 

Let's practice.

162 =____

 

 


202 = ___

 



You can also square a decimal or fraction.

 

 

Square Roots

Let's practice
What the square root of 13 to the nearest hundredth?

 

 

 



Parts of a Right Triangle

 

 

 

 

 

 

Finding the Hypotenuse

 

 

 


Let’s Practice.

Find the length of the unknown side. Round to the nearest 100th.

 

 

 

 


Finding A Leg

 

 




Let’s Practice

Find the length of the unknown side. Round to the nearest 100th.

 

Solution:

 

 

Determine if a triangle is a right triangle.

To determine if a triangle is a right triangle, when you know 3 sides, see if a2 + b2 = c2.
 

A = 40, b = 9, c = 41

a2 + b2 = c2

402 + 92 = 412

1600 + 81= 1681

Since this is true, you have a right triangle.

 

 


Modeling in Real Life

Hayden is building a shed. He measures part of the corner frame and finds it 8 feet going up and 6 feet across the bottom. The diagonal is 10 feet. Is it a right angle? In other words, will his corner be square?


The Pythagorean Theorem can be used to see if the corner is at a right (90 degrees) angle.


If 8 squared plus 6 squared is equal to 10 squared then it is a right angle.


Since 82 + 62= 100 and 102 = 100 therefore the corner is square.