The Pythagorean Theorem

x

z y

𝑥2+𝑦2=𝑧2

The Pythagorean Theorem

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 TheoremCopy the labels to the corresponding sides

a

b

c

a

a

a

b

b

b

c

c

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

The Pythagorean Theorem

+ can also be written as:

(Small leg)2 + (Large leg)2 = hypotenuse2

a

b

c

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

Example 2Again We Solve the Equation

Given

Simplify powers

65 Subtraction property

x = Square root both sides

x = 8.062 Square root of 65