Obj. 6 The Pythagorean Theorem

Objectives

Use the Pythagorean Theorem to solve problems.

The Pythagorean Theorem (a2 + b2 = c2) states the relationship between the sides of a right triangle. Although it was named for Pythagoras (circa 500 B.C.), this relationship was actually known to earlier people, including the Babylonians, Egyptians, and the Chinese.

A Babylonian tablet from 1800 B.C. listing sides of right triangles.

The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: the Remember: the Remember: the Remember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.

x

12

13

The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: the Remember: the Remember: the Remember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.

x2 + 122 = 132x

12

13

The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: the Remember: the Remember: the Remember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.

x2 + 122 = 132

x2 + 144 = 169x

12

13

The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: the Remember: the Remember: the Remember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.

x2 + 122 = 132

x2 + 144 = 169

x2 = 25

x

12

13

The Pythagorean Theorem allows us to find an unknown side of a right triangle if we know the other two sides. Remember: the Remember: the Remember: the Remember: the hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.hypotenuse is always c.

x2 + 122 = 132

x2 + 144 = 169

x2 = 25

x = 5

x

12

13

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =xxxx ----2222

xxxx x2 -2x

----2222 -2x 4

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =xxxx ----2222

xxxx x2 -2x

----2222 -2x 4

2 216 x 4x 4 x+ + =

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =xxxx ----2222

xxxx x2 -2x

----2222 -2x 4

2 216 x 4x 4 x+ + =

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =xxxx ----2222

xxxx x2 -2x

----2222 -2x 4

2 216 x 4x 4 x+ + =20 4x = 0

2

6

x

x x-2

4

Examples Find the value of x. Reduce radicals to simplest form.

1.

2.

2 2 22 6 x+ =24 36 x+ =

240 x=

x 2 10=

2 2 24 (x 2) x+ =xxxx ----2222

xxxx x2 -2x

----2222 -2x 4

2 216 x 4x 4 x+ + =20 4x = 0

20 = 4xx = 5

2

6

x

x x-2

4

Pythagorean Triple

A set of nonzero whole numbers a, b, and c, such that a2 + b2 = c2.

Memorize these!

Note: 3, 4, 5 is the onlyonlyonlyonly triple that contains three consecutive numbers.

Pythagorean TriplesPythagorean TriplesPythagorean TriplesPythagorean Triples

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

x2x2x2x2 6, 8, 10 10, 24, 26 14, 48, 50 16, 30, 34

x3x3x3x3 9, 12, 15

x4x4x4x4 12, 16, 20

x5x5x5x5 15, 20, 25

Examples Find the missing side of the right triangle.

1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

Examples Find the missing side of the right triangle.

1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

Examples Find the missing side of the right triangle.

1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212

Examples Find the missing side of the right triangle.

1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212

5555

Examples Find the missing side of the right triangle.

1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212

5555

17171717