4.11.1 Pythagorean Theorem

4.11.1 The Pythagorean Theorem

The student is able to (I can):

Use the Pythagorean Theorem to solve problems.

Use Pythagorean inequalities to classify triangles.

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


x2 + 122 = 132x

12

13


x2 + 122 = 132

x2 + 144 = 169x

12

13


x2 + 122 = 132

x2 + 144 = 169

x2 = 25

x

12

13


x2 + 122 = 132

x2 + 144 = 169

x2 = 25

x = 5

x

12

13

Square Roots

When we are taking the square root of a number, we will not always get a whole number answer.

If your answer is not a whole number, then the number your calculator gives you is a decimal approximationapproximationapproximationapproximation. This number is like , it goes on forever.

If I ask for an exact answerexact answerexact answerexact answer, I do notnotnotnotwant a decimal I want you to leave it as a simplified radicalsimplified radicalsimplified radicalsimplified radical.

To simplify a radical (square root):

Find all the prime factors of the number

Group pairs of factors these can be pulled out of the radical

Any factors that cannot be paired up must stay inside the radical

Example: Simplify 24

24

2 12

2 6

2222 3333

=i2 2 3 2 6

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

1.

2.

2

6

x

x x-2

4


1.

2.

2 2 22 6 x+ =2

6

x

x x-2

4


1.

2.

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

2

6

x

x x-2

4


1.

2.

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

240 x=

2

6

x

x x-2

4


1.

2.

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

240 x=

x 2 10=

2

6

x

x x-2

4


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


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


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


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


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 = 020 = 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, ____


1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555


1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212


1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212

5555


1. 3, 4, ____

2. 9, ____, 15

3. ____, 12, 13

4. 8, 15, ____

5555

12121212

5555

17171717

Thm 5-7-1

Thm 5-7-2

Converse of the Pythagorean Theorem

If a2 + b2 = c2, then the triangle is a right triangle.

Pythagorean Inequalities Theorem

If then the triangle is an obtuseobtuseobtuseobtuse triangle.

If then the triangle is an acuteacuteacuteacute triangle.

2 2 2c a b ,> +

2 2 2c a b ,< +

Classifying Triangles

right triangleobtuse triangle

acute triangle

> +2 2 2c a b < +2 2 2c a b= +2 2 2c a b

a

b

c

a

b

ca

b

c

Examples Classify the following triangle measures as right, obtuse, or acute.

1. 5, 7, 10

2. 16, 30, 34

3. 3.8, 4.1, 5.2

Examples Classify the following triangle measures as right, obtuse, or acute.

1. 5, 7, 10

102 = 100, 52 + 72 = 74 ObtuseObtuseObtuseObtuse

2. 16, 30, 34

342 = 1156, 162 + 302 = 1156 RightRightRightRight

3. 3.8, 4.1, 5.2

5.22 = 27.04, 3.82 + 4.12 = 31.25 AcuteAcuteAcuteAcute

4.11.1 Pythagorean Theorem

Education

Transcript of 4.11.1 Pythagorean Theorem