The Pythagorean Theorem
For this proof we must draw ANY right Triangle:
Label the Legs “a” and “b” and the hypotenuse “c”
a
b
c
The Pythagorean TheoremCopy the triangle three more times and build a
square with each triangle at a corner
a
b
c
The Pythagorean Theorem
a
b
c
a
a
a
b
b
b
c
c
c
The Area of the big square can be found by multiplying the lengths of the sides (a + b)(a + b)
The Pythagorean Theorem
a
b
c
a
a
a
b
b
b
c
c
c
The Area of the big square can also be found by adding the 4 triangles and the smaller inner square:
4(ab) +
The Pythagorean Theorem
a
b
c
a
a
b
b
c
c
c
Since both equations find the area of the same object, they must be equal:
4(ab) + = (a + b)(a + b)
The Pythagorean TheoremNow we need to do some simplifying:
4( ) + = ()()
2 + = + 2 + Multiply Polynomials
= + Subtract 2
Since we began with any random right triangle, the statement
+ will work for EVERY right triangle
Example 1For this example we will be given both legs and will be trying to find the hypotenuse
10
x
6
Example 1Setup the equation using the
Pythagorean Theorem: (Small leg)2 + (Large leg)2 = hypotenuse2
10
x
6
𝟔𝟐+𝟏𝟎𝟐=𝒙𝟐
Example 1Now Solve the Equation
Given
Simplify powers
136 Addition property
= x Square root both sides
11.662 = x Square root of 136
Example 2For this example we will be given a leg and
the hypotenuse and will be trying to find the other leg
9
x4
Example 2
9
x4
Setup the equation using the Pythagorean Theorem:
(Small leg)2 + (Large leg)2 = hypotenuse2
42 + x2 = 92