2.4  Pythagorean Theorem 

 

Pythagorean Theorem:

In any Right Triangle with vertices A, B, & C the following will be true:
  

  • Triangle sides a and b will be the Legs of the Right Triangle
        
  • Triangle side c will be the Hypotenuse of the Right Triangle
       
  • We get the following relationship:
         

Example A

Question:  Why is everything squared?


 One Possible Solution:

Note: Do not label any points

Step:
1.  Create a Square ( Regular Polygon )
2.  On the Left Side of this Square, create a
      line through the 2-points on the left side.
3.  Create a Point on this line above the Square
4.  Create a Square using the top left point of
      the Square and the point made in Step 3
5.  Create a Square using bottom left corner
     of each Square
6.  Use the Area tool to find the area of
     each Square
7.  Add the area of the 2-smaller squares
     together, and you will find that they total up to
     the area of the large square ( within an
     acceptable error for rounding. )

The 3-squares create a Right Triangle and the sides of the triangle make up a side on a square and the area of a square is found by multiplying 2-sides of a square together or a Side-Squared.

Press or to run the Geogebra build simulation.

Watch the video for this example:

Example B

Problem:

Given ΔABC where C = 90o , a = 3, & b = 4.
Find the length of "c" the hypotenuse.

 

                  

Answer:



So now we know that the length of the hypotenuse is 5.


Watch the video:

 

Quick Question...

To use the app:

  • Click the problem type you want.
    or or or
  • Clickto begin, click for full screen mode
  • Click to check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

Instructions:

  • Click the check box next to the variable
    where you want to enter a value
  • Click in the textbox to enter your answer.
  • Repeat as often as needed.
  

Example C

Problem:

Given ΔABC where C = 90o , a = 3, & c = 5.
Find the length of side "b"


 Answer:



So now we know that the length of side b is 4.


Watch the video:

Quick Question...

To use the app:

  • Click the problem type you want.
    or or or
  • Clickto begin, click for full screen mode
  • Click to check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

Instructions:

  • Click the check box next to the variable
    where you want to enter a value
  • Click in the textbox to enter your answer.
  • Repeat as often as needed.
  

 

So, what happens if a2 + b2 ≠ c2  ?

As long as a, b, & c form a triangle, we get this...

Pythagorean Inequality Theorem:

In  ΔABC  where sides a and b are the two shorter sides and side c is the longest side.

  • If a2 + b2 < c2     then the triangle is Obtuse.
        
  • If a2 + b2 > c2     then the triangle is Acute.
      
       
 

Example D

Question:

Use Geogbra to verify your calculations for each triangle

  • Given ΔABC where a = 7, b = 6, & c = 11, is this triangle Acute, Obtuse, or Right?
        
  • Given ΔABC where a = 8, b = 9, & c = 12, is this triangle Acute, Obtuse, or Right?

Answer:

If we try to make a triangle with these dimension we get...              ...an Obtuse triangle



If we try to make a triangle with these dimension we get...                  ...an Acute triangle


Watch the video:

 

Now let's try that with a triangle made with 3-squares...


 Note: Do not label any points

Step:
1.  Create a Square ( Regular Polygon )
2.  Create a Square using the top right point of
      the Square and the point made in Step 1
3.  Create a Square using bottom right corner
     of each Square
4. Use the Area tool to find the area of
     each Square
5.  Adjust the position of the squares to make an
     obtuse triangle bordered by the 3-squares.
6.  Repeat steps 1-4 and then Adjust the position
     of the squares to make an acute triangle
      bordered by the 3-squares.

Now use Geogebra ot verify the Pythagorean Inequality Theorem.

Press or to run the Geogebra build simulation.

... and lastly we have...

Pythagorean Triples:

A Pythagorean triple is a set of three nonzero whole numbers a, b, and c
such that a2 + b2 = c2    

Two of the most well-known sets of Pythagorean triples are (3, 4, 5) and  (5, 12, 13).

An easy way to find Pythagorean triples is to multiply one of  these two sets by a whole number.

For example, multiplying the first set by 2 yields (6, 8, 10), which is also a Pythagorean triple.

16 Primitive  Pythagorean Triples :
( not multiples of another Pythagorean Triple )

(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)

(20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)

(11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)

(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

 

Practice Problems


A.  Given ΔABC where C = 90o , a = 6, & b = 14.
      Find the length of "c" the hypotenuse rounded to the nearest tenth.
      		 
B.  Given ΔABC where C = 90o , a = 2, & c = 20.
      Find the length of side "b" rounded to the nearest tenth.
   
C.  Given ΔABC where C = 90o , a = 40, & c = 41.
      Find the length of side "b" rounded to the nearest tenth.
     		 
D.  Given ΔABC where a = 4, b = 6, & c = 11, is this triangle Acute, Obtuse, or Right?
  		 
E.  Given ΔABC where a = 6, b = 6, & c = 7, is this triangle Acute, Obtuse, or Right?
 
 
F.  Identify one Pythagorean Triple that is not listed in the lesson.  

 

Assignment:

Instructions:

  • Before you do each problem:
    Screenshot the app window to show the question,
    then paste it into the Jamboard for the problem#
  • While you are doing the problem:
    Show all work needed for each  problem.
  • After you have done the problem:
    Screenshot the app window to show the question and score,
    then paste it into the Jamboard next to the work you did.

 

Pythagorean Theorem:  

To use the app:

  • Click the problem type you want.
    or or or
  • Clickto begin, click for full screen mode
  • Click to check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

Instructions:

  • Click the check box next to the variable
    where you want to enter a value
  • Click in the textbox to enter your answer.
  • Repeat as often as needed.
  

Problems 1 - 6,

Find the Hypotenuse, use:

Problems 7 - 12,

Find Leg A, use:

Problems 13 - 18,

Find Leg B, use:

Problems 19 - 24,

Find the Missing Side, use: