# Pythagorean Theorem

See how to use the pythagorean theorem

to find the side of a right triangle.

5 examples and their solutions.

## Pythagorean Theorem

### Theorem

a

^{2}+ b

^{2}= c

^{2}

(a + b)

^{2}= 4⋅12ab + c

^{2}- [4]

a

^{2}+ 2ab + b

^{2}= 2ab + c

^{2}- [5]

∴ a

^{2}+ b

^{2}= c

^{2}

[1]

Draw a right triangle like this.

[2]

Use the above right triangle

to make a square like this.

to make a square like this.

[3]

m∠[plane] + m∠[dot] + 90 = 180

So the interior angles of the middle quadrilateral

are all 90°.

So the middle quadrilateral is a square.

So the interior angles of the middle quadrilateral

are all 90°.

So the middle quadrilateral is a square.

[5]

Close

### Example

### Example

x

^{2}+ 5

^{2}= (√89)

^{2}

x

^{2}+ 25 = 89

x

^{2}= 64

x = 8

Close

## Pythagorean Triples

### Definition

3, 4, 5

5, 12, 13

7, 24, 25

...

The Pythagorean triple is three positive integers5, 12, 13

7, 24, 25

...

that satisfy the Pythagorean theorem:

a

^{2}+ b

^{2}= c

^{2}.

In high school,

these triples (and their multiples) are mostly used.

### Example

x = 3

(x, 4, 5)

→ (3, 4, 5)

→ x = 3

→ (3, 4, 5)

→ x = 3

Close

### Example

x = 13

(5, 12, x)

→ (5, 12, 13)

→ x = 13

→ (5, 12, 13)

→ x = 13

Close

### Example

x4 = 63 - [2]

x4 = 2

x = 8

[1]

(6, x, 10)

→ ×2 of (3, 4, 5)

→ So draw (3, 4, 5) right triangle

next to the given triangle.

→ ×2 of (3, 4, 5)

→ So draw (3, 4, 5) right triangle

next to the given triangle.

[2]

Close