Lesson 10-1: The Pythagorean Theorem

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LESSON 10-1: THE PYTHAGOREAN THEOREM

SIMPLIFYING RADICALS LESSON 10-2

ESSENTIAL UNDERSTANDING A radical is any number with a square root. You can simplify a radical expressions using multiplication and division.

MULIPLICATION PROPERTY OF SQUARE ROOTS: √32 = √16 ∙ √2 = 4√2

PROBLEM 1What is the simplified form of √160?Ask “what perfect square goes into 160?”

160 = 2 ∙ 80 No perfect square160 = 4 ∙ 40 Yes perfect square160 = 16 ∙ 10 Yes perfect square

√160 = √16 ∙ √10 = 4√10

GOT IT? 1Simplify: √72

6 √2

PROBLEM 2 Simplify: √54n7

√9 ∙ 6 ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n√9 ∙ √6 ∙ √n2 ∙ √n2 ∙ √n2 ∙ √n

3 ∙ √6 ∙ n ∙ n ∙ n ∙ √n3n3 ∙ √6n3n3√6n

GOT IT? 2Simplify: -m √80m9

PROBLEM 3Simplify: 2√7t ∙ 3√14t2

2 ∙ 3 ∙ √ 7t ∙ 14t2

6 ∙ √98t3

6 ∙ √49t2 ∙ 2t6 ∙ 7t ∙ √2t

42t√2t

DIVISION PROPERTY OF SQUARE ROOT

√3649 √ 144

PROBLEM 5

√8x3

50x8x3 = √4 2 x2 x50x = √25 2 x

RATIONALIZING THE DENOMINATOR

It’s okay to have a square root in the numerator, but not the denominator. It’s not simplified enough if you keep a square root in the denominator.

√3√7

√7√7

√21√49

√21 7

Really equals 1

OPERATIONS WITH RADICAL EXPRESSIONSLESSON 10-3

COMBINING “LIKE” RADICALS

3√5 and 7√5 have the same radicand.

Radicand = number under the square root.

-2√9 and 4√3 do not have the same radicand.

If two or more numbers have the same radicand, then we can combine them together.

PROBLEM 1What is the simplified form of 2√11 + 5√11?

2√11 + 5√11We could break it down even more…

(√11 + √11) + (√11 + √11 + √11 + √11 + √11)How many √11’s do we have altogether?

7 √11

2√11 + 5√11 = 7 √11

WHAT IS THE SIMPLIFIED FORM OF √3 - 5√3?

√3 - 5√31√3 - 5√3(1 – 5)√3

-4√3Got it?

1. 7√2 - 8√2 2. 5√5 + 6√5

PROBLEM 2: WHAT IF THEY DON’T “LOOK LIKE THEY CAN BE SIMPLIFIED?

5√3 - √12Simplify √12 to see if there is a perfect

square.√12 = √4 ∙ 3 = 2√3

So we have 5√3 - 2√3.

5√3 - 2√3 = 3√3

GOT IT? 2

1. 4 √7 + 2 √28

2. 5 √32 - 4 √18

PROBLEM 3: USING THE DISTRIBUTIVE PROPERTY

10(6 + 3) = 10(6) + 10(3) = 60 + 30 = 90

In the same way…√10(√6 + 3)

Use the Distributive Property(√10 ∙ √6) + (√10 ∙ 3)

√60 + 3√10Break down 60 to find a perfect square.

√4 ∙ √15 + 3√102 √15 + 3 √10

Can we simplify even more?

PROBLEM 3: (√6 - 2 √3)(√6 + √3)

(√6 - 2 √3)(√6 + √3)Carefully FOIL

(√6)(√6) + (- 2 √3)(√6) + (√6)(√3) + (- 2 √3)(√3) First Inside Outside Last √36 + -2√3 ∙ 6 + √6 ∙ 3 + -2√3 ∙ 3 6 + -2√18 + √18 + -2 ∙ 3

6 – √18 – 6-√18 = -1 √9 ∙ 2

= -1 ∙ 3 √2 = -3√2

GOT IT? 31. √2(√6 + 5)

GOT IT? 32. (√11 – 2)2

GOT IT? 33. (√6 – 2 √3)(4 √3 + 3 √6)

PROBLEM 4: COJUGATESExamples:

+ and - + 8 and - 8

What do you notice?Cojugates: the sum and difference of the same

two terms. ( + )( - )7 – 3 = 4

PROBLEM 4: RATIONALIZING A DENOMINATOR

Multiply by the cojugates. =

= = 2 + 2

SOLVING RADICAL EQUATIONSLESSON 10-4

ESSENTIAL UNDERSTANDINGRadical equations = equations with a radicand (square root)

Some radical equations can be solved by squaring each side.

The expression under the radicand MUST be positive.

PROBLEM 1What is the solution of 7 + = 16?

7 + = 16Subtract 7 from both sides.

= 9Square both sides.

()2 = 92

x = 81

GOT IT? 1What is the solution of -5 + = -2?

PROBLEM 2In the expression, t = 2, what is m when t is 3?

3 = 21.5 =

1.52 = ()2

2.25 =

(2.25)(3.3) = (3.3)

7.425 = m

PROBLEM 3What is the solution of ?

()2 = ()2

5t – 11 = t + 54t – 11 = 5

4t = 16t = 4

GOT IT? 3

EXTRANEOUS SOLUTIONSTake the original equation x = 3.

Let’s square each side.

x2 = 9

The solutions would be 3 and -3….right?

However, -3 doesn’t fit in our original equation.

Sometimes when we square each sides, we create a false solution.

n2 = n + 12n2 - n – 12 = 0

(n – 4)(n + 3) = 0n = 4 and -3

Does both numbers work for n?

PROBLEM 4

(n – 4)(n + 3) = 0n = 4 and -3

PROBLEM 4

PROBLEM 5What is the solution √3y + 8 = 2?

√3y = -6Can you ever have a negative as a product of a square

root?(√3y)2 = -62

3y = 36y = 12

Check:

LESSON CHECK 1

LESSON CHECK 2

LESSON CHECK 3

LESSON CHECK 4

GRAPHING SQUARE ROOT FUNCTIONSLESSON 10-5

SQUARE ROOT FUNCTION IS…A function where the “x” or independent

variable is under the square root.

The parent square root function is y = .

WHAT IS THE DOMAIN AND RANGE?

Domain: What numbers can you put in for x?

All positive numbers

Range: What kind of numbers are the output of y?

All positive numbers

PROBLEM 1What is the domain and range of this function?

y = 2Domain: what numbers can you put in for x?

*the radicand can not be negative*

3x – 9 03x 9x 3

So, the domain of this function are all real numbers greater or equal to 3. Range: all positive numbers

GOT IT? 1What is the domain of y = ?

PROBLEM 2Graph the function I = , which gives the current I in ampere for a certain circuit with P watts of power. When will the current exceed 2 amperes?

GOT IT? 2When will the current exceed 1.5 amperes?

PROBLEM 3Take the parent square root function. y =

If you want to move it UP on the graph, you add OUTSIDE the square root.

If you want to move it DOWN on the graph, you subtract OUTSIDE the square root.

PROBLEM 4Take the parent square root function. y =

If you want to move it RIGHT on the graph, you SUBTRACT INSIDE the square root.

If you want to move it LEFT on the graph, you ADD INSIDE the square root.

GOT IT? 3 AND 4What coordinate does the graph of y = start on?

What coordinate does the graph of y = start on?

What coordinate does the graph of y = 4 start on?

LESSON QUIZ1. Is y = x a square root function? Why or why not?

2. Can a domain of a square root function include negative numbers? If it can, give an example. If it can not, explain.

HOME WORK#8 – 38 evens

On 16 – 24 just make a table.

On 30 – 38, tell me the coordinate that the graph will start. You don’t need to graph it.

TRIGONOMETRIC RATIOSLESSON 10-6

KEY CONCEPT: TRIG RATIOS

KEY CONCEPT: TRIG RATIOSOr…

Sine A =

Cosine A =

Tangent A =

EXAMPLEWhat are the sine of R?

Sine R =

EXAMPLEWhat are the cosine of R?

Cosine R =

EXAMPLEWhat are the tangent of R?

Tangent R =

GOT IT? 1What is the sine, cosine and tangent of E?

PROBLEM 2What is the cosine of 55 degrees?Step 1: Make sure your calculator is in Degree mode.

Step 2: Press “cos” and then 55 and then “enter”

GOT IT? 2Use a calculator to compute these trigonometric ratios.1. Sin 80

2. Tan 45

3. Cos 15

4. Sin 9

PROBLEM 3We can use sine, cosine or tangent to solve for x.

How does 14, 48 and x relate to each other?What ratio connects all three together?

Sine 48 =

Sin 48 =

Multiply each side by 14. 14 Sin 48 = x

x ≈ 10.4

GOT IT? 3To the nearest tenth, what is the value of x in the triangle?

FLIP CHART

INVERSE OF TRIG RATIOSSine(Sine-1) = 1 Cosine(Cosine-1) = 1

Tangent(Tangent-1) = 1

Sine and Sine-1 are inverses of each other.

Cos(Cos-1)(45) = 45

PROBLEM 4Find the angle of A in the triangle.

GOT IT? 4In a right triangle, the side opposite angle A is 8mm and the hypotenuse is 12 mm long. What is the angle of A?

OUTSIDE ANGLESAngle of Elevation: angle from the horizontal UP to the line of sight.

Angle of Depression: angle from the horizontal DOWN to the line of sight.

PROBLEM 5Suppose you are waiting in line for a ride. You see your friend at the top of the ride. How fare are you from the base of the ride?

PROBLEM 5

Tan 20 =

x tan 20 = 150

x = 410

You are about 410 feet from the base of the ride.

HOME WORK#31 – 35, 39 – 42 all

Lesson 10-1: The Pythagorean Theorem

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