TODAY

y = mx + b

Review for Radical Test tomorrow

= 2

2x 4 x 7 0.

Set up the equation so that

there will be one radical on

each side of the equal sign.

2x 4 x 7

Square both sides.

2x 4 2

x 7 2

Simplify.

2x + 4 = x + 7

x = 3

Verify your solution.

2x 4 x 7 0Therefore, the

solution is

x = 3.

Solve

Solving Radical Equations

2(3) 4 3 7

10 10

0

L.S. R.S.

0)105()23( xx

)105()23( xx

22

)105()23( xx

10523 xx

1022 x

x212

Isolate the radicals

Square both sides

Simplify

Subtract 3x from both sides

Add 10 to both sides

When there are two radicals on the same side of an equation, isolate both radicals by moving one to the other side of the equal sign.

x = 6

final three...

The Pythagorean Theorem

Pythagoras of Samos

• Lived in southern Italy from 571 BC-495 BC

• Considered the first true mathematician

• Used mathematics as a means to understand the natural world

• First to teach that the earth was a sphere that revolves around the sun

The Pythagorean Theorem

8"

6"

x"

The Pythagorean Theorem

The Pythagorean Theorem

The Pythagorean Theorem

484 = 81 + 196 ?

The Pythagorean Theorem

21"

x"

The Pythagorean Theorem

18’9”

The Pythagorean Theorem

• Solution: x2 = 902 + 902 = 16,200 x = 127.28 ft

Class Work: 4.9

Common Pythagorean Triples

6 8 10

5 12 137 24 25

8 15 17

3 4 5

a2 + b2 = c2

12 16 20

In a right triangle, the side opposite the right angle is the longest side. It is the hypotenuse. The other two sides are the legs of a right triangle.

legs

hypotenuse

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2a

b

c

A right triangle has sides of lengths 20, 29,

and 21. What is the length of the

hypotenuse?

Verify that the Pythagorean Theorem is true

for the right triangle in the previous

question.

Find the length of the hypotenuse of a right

triangle with legs of lengths 7 and 24.

WHO? Czech-American mathematician

Olga Taussky-Todd (1906-1995) studied

Pythagorean triangles. In 1970, she won

the Ford Prize for her research

Find the value of x. Leave your answer in

simplest radical form.

The hypotenuse of a right triangle has length

12. one leg has length 6. Find the length of

the other leg in simplest radical form.

820

x

1. 2.

3. 4.

x

8

6

x

x

x x8

x26 26

48

2x

3

3

When the lengths of the sides of a right

triangle are integers, the integers form a

Pythagorean Theorem. Here are some

common primitive Pythagorean Triples.

3, 4, 5

5, 12, 13

8, 15, 17

7, 24, 25

Choose an integer. Multiply each number of

a Pythagorean triple by that integer. Verify

that the result is a Pythagorean triple.

9, 40, 41

11, 60, 61

12, 35, 37

13, 84, 85

What is the length of the diagonal of a rectangle whose sides measures 5 and 7?

Calculate the length of the side of a square whose diagonal measures 9 cm.

What is the measure of the longest stick we can put inside a 3 cm x 4 cm x 5 cm box?

In ΔABC with longest side c,

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

if c2 > a2 + b2, then the triangle is obtuse.

if c2 < a2 + b2, then the triangle is acute.

B

C A

a

b

c

a) 2, 3, 4

b) 3, 4, 5

c) 4, 5, 6

d) 3, 3, 3 2

e) 3, 3, 3 3

f) 2, 2 3, 4

g) 5, 5, 5

h) 4, 4, 5

i) 2, 2, 2

j) 2.5, 6, 6.5

The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.

obtuse

right

acute

right

obtuse

right

acute/equi

acute

right

right

a) 2, 3, 4

b) 3, 4, 5

c) 4, 5, 6

d) 3, 3, 3 2

e) 3, 3, 3 3

f) 2, 2 3, 4

g) 5, 5, 5

h) 4, 4, 5

i) 2, 2, 2

j) 2.5, 6, 6.5

The number represent the lengths of the sides of a triangle (a, b, c). Classify each triangle as acute, obtuse, or right.

obtuse

right

acute

right

obtuse

The Pythagorean Theorem

Proof of the Pythagorean Theorem:

23 x 123(2)2(2)+3(1)2(1)

672= 276

18 x 378(7)

624 + 7+ 5

6 3+ 3

6= 666

WorkBackward

Variables represent numbers that are unknown at the time. Even

after the variables are known, all of the rules of algebra still apply.

For example, let's foil these two terms:

60"36"

x