The Pythagorean

Theorem a² + b² = c²

Pythagoras's Theory diagram

Pythagoras’s theorem

Who Invented It?Who Invented It?

Pythagoras was a Greek Pythagoras was a Greek mathematician mathematician

He created the theorem (which was of He created the theorem (which was of course named after him)course named after him)

Pythagoras’s theory is “In any right-Pythagoras’s theory is “In any right-angle triangle, the square of the angle triangle, the square of the hypotenuse is equal to the sum of two hypotenuse is equal to the sum of two squares on the other two sides. (asquares on the other two sides. (a² + ² + b² = c²)”b² = c²)”

(Pythagoras)

Right Angle Triangle A right angle triangle is a triangle that has one

right angle Every right angle triangle is labelled the same

way. The side opposite the right angle is called the hypotenuse and is always labelled ‘c’.

(Right Angle Triangle)

The other two sides are labelled ‘a’ for the altitude and ‘b’ for the base. (The ‘a’ and ‘b’ sides can also be called a leg)

An example of finding a An example of finding a missing Hypotenusemissing Hypotenuse

The Pythagorean theorem is a way of calculating the sides of a right angle triangle

3

4

?Ex. for the Pythagorean Theorem: say square A is 3 x 3 and the B square is 4 x 4 (The ex. Triangle with the measurements for the sides ‘a’

and ‘b’)

The Pythagorean TheoremThe Pythagorean Theorem

You already have the measurements of the altitude and the base side squares and you already have one equal square on each side of the triangle and so you begin to solve the Pythagorean Theorem

3

4

?

Continuing the example of Continuing the example of finding the missing finding the missing

HypotenuseHypotenuse Next step to finding the hypotenuse is Next step to finding the hypotenuse is

to put the two A and B square numbers to put the two A and B square numbers into the theory/equation (ainto the theory/equation (a² + b² = c², ² + b² = c², into into

3² + 4² = c²)3² + 4² = c²) Then solve (in this case) Then solve (in this case) the 3² and 4² part which the 3² and 4² part which would then equal 9 and would then equal 9 and 16 (9 + 16 16 (9 + 16 = c²) = c²)

9

16

? Still a mystery

(Again the diagram of right angle triangle and the three equal triangles with A and B numbers filled in)

Still Continuing the example Still Continuing the example of finding the missing of finding the missing

HypotenuseHypotenuse And then add 9 and 16 which makes And then add 9 and 16 which makes

25 and then because there is still a ² 25 and then because there is still a ² on the c you have to square root it ( on the c you have to square root it ( ) or find what multiplies equally ) or find what multiplies equally twice into the sum (ex. 25) twice into the sum (ex. 25)

The answer to the example would be The answer to the example would be 5 (5 x 5 = 25)5 (5 x 5 = 25)

The hypotenuse is now solvedThe hypotenuse is now solved 53

4(The Pythagorean diagram with the solved question)

An example now of finding a An example now of finding a missing legmissing leg

Finding the missing leg is like finding Finding the missing leg is like finding the missing hypotenuse but some of the missing hypotenuse but some of the steps are differentthe steps are different

When you begin you already have When you begin you already have the measurements for the the measurements for the hypotenuse and either the base or hypotenuse and either the base or the altitude side and you again have the altitude side and you again have one equal square on each side one equal square on each side of the triangleof the triangle

5 5

44

??

In this case the hypotenuse is 5 x 5 and the base is 4 x 4 and we do not know the altitude(The ex. Triangle with the measurements for sides ‘c’

and ‘b’)

Continuing the example of Continuing the example of finding a missing legfinding a missing leg

Next step is also the same as last Next step is also the same as last time where you put the C and either time where you put the C and either A/B (whichever one was already A/B (whichever one was already found) into the equation (afound) into the equation (a² + b² = ² + b² = c², and this time it turns into a² + 4² c², and this time it turns into a² + 4² = 5²)= 5²)

Then solve (in this case) the 4² and Then solve (in this case) the 4² and 5² part which would then be (again 5² part which would then be (again in this case) 16 and 25 in this case) 16 and 25

25

16

?The diagram of the Pythagoras right angle triangle with the area filled in for squares C and B and A still unknown

StillStill Continuing the Continuing the example of finding a example of finding a missing legmissing leg Here where you go a different way Here where you go a different way

then finding the missing hypotenusethen finding the missing hypotenuse Now you take aNow you take a² + 16 = 25 and ² + 16 = 25 and

change the problem around until the change the problem around until the aa² is by itself on one side (kind of like ² is by itself on one side (kind of like algebra) and turn it into algebra) and turn it into aa² = 25 – 16² = 25 – 16

Next solve 25 – 16 (equals 9) and then Next solve 25 – 16 (equals 9) and then figure out what multiplies by itself figure out what multiplies by itself equally into 9 or square root it ( ) equally into 9 or square root it ( ) and this time the answer is 3 (3 x and this time the answer is 3 (3 x 3 = 9)3 = 9)

The leg is solvedThe leg is solved

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5

(The finished problem)