Today’s Class• Do now:

– Work on Warm UP– Get out HW

• Objective– SWBAT apply the Pythagorean theorem to solve for missing side

lengths– SWBAT apply the converse of the Pythagorean theorem to

classify a triangle by its angles– SWBAT to apply Pythagorean formula to distance problems

• Homework: Problem Set, Corrections, Binder

Announcements

Everyone needs a Binder by Friday- Three rings- Only for geometry

Retakes due Friday at 5:00pmQuiz Friday

We will learn to…

• Apply the Pythagorean theorem to solve for missing side lengths

• apply Pythagorean formula to distance problems

Background Vocabulary

radical

This is also called a “Square Root” radicand

Steps to Simplify Radicals1. Find the largest __________________ of the

radicand, excluding 1.Note: A factor is a number that divides evenly into a given number with no remainder.

2. Write the number as the _______ of the perfect square factor and its other factor.

3. Split each factor into two separate _______.

4. Simplify the _____________ radical.

perfect square factor

product

radicals

perfect square

Example 1: 1. Find the largest perfect square factor of 75, excluding 1.

75

2. Write the radicand as the product of the perfect square factor and its corresponding factor.

75

Check for perfect square factors of 75: 4, 16, 25, 36, 49, 64

325

Example 1: 3. Split each factor into two separate radicals.

75

4. Simplify the perfect square radical.

325

35

325

35

Example 2: • First Step: Multiply the radicals

• Then…try to simplify on your own• Lastly, check if you have the same answer as the

others at your table.

24 2√3

More Simplification…

• What if the number under the radical is a fraction?

Example 3: Rationalizing the Denominator

163

163

2. Simplify each radical if possible.

43

Is this radical in simplest form?

No, we cannot have a radical in the denominator.

1. Separate the numerator and denominator into separate radicals.

We must rationalize the denominator.

Example 3: Rationalizing the Denominator

4. Simplify.

43

3. Multiply both the numerator and denominator by the root.

Why can we do this without changing the result?

4 33

33

Example 4: Rationalizing the Denominator

272

272

932

9 32

3 32

Simplify the radical.

Rationalize the denominator.

3 32

22

3 62

The values in the chart represent the sides of a right triangle. Complete the chart below.

•Compare the values of a2 + b2 and c2. Write an algebraic equation to represent this relationship.

•Describe what the variables in your equation above represent by completing the following phrases:

- ∆XYZ is a _____________ triangle because ___________________________________________________________

- a and b represent the _____________ of ∆XYZ because ________________________________________________

- c represents the __________________ of ∆XYZ because ________________________________________________

9 16 25 25

25 144 169 169

.36 .64 1 1

LEGS they form the right angle of the triangle

HYPOTENUSE

c2 = a2 + b2

right it has a box (right angle)

it is opposite the right angle of the triangle

Pythagorean Theorem: If a triangle is a _____________ triangle,

then the square of the longest side (______________) is equal to the

____________of the squares of the other two sides (__________).

RIGHT

HYPOTENUSE

SUM LEGS

OR

How does this diagram represent Pythagorean Theorem?

Example 1a

Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth.

SOLUTION

(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem

52 = 32 + x2

25= 9 + x2

16 = x2

4 = x Find the positive square root.

Substitute.

Square.

Subtract 9 from both sides

Example 1a

Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth.

SOLUTION

Not possible because not a right triangle

Example 1c

SOLUTION

(hypotenuse)2 = (leg)2 + (leg)2Pythagorean Theorem

x2 = 4 + 12

x2 = 16

Find the positive square root.

Substitute.

Square.

Add.

x = 4

Find the value of x. Write your answer in simplest radical form and as a decimal rounded to nearest hundredth.

2

Example 1c

Maritza and Melanie run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest hundredth of a mile, they must travel to return to their starting point?

EXAMPLE 4

SOLUTION

= +

162 = 42 + x2

EXAMPLE 3 Standardized Test Practice

Find positive square root.

Substitute.

Square.

Subtract 16 from each side.

SOLUTION

Approximate with a calculator.

162 = 42 + x2

256 = 16 + x2

15.491 ≈ x

240 = x

240 = x2

ANSWER

The ladder is resting against the house at about 15.5 feet above the ground.

The correct answer is D.

Example 4

SOLUTION

(hypotenuse)2 ? (leg)2 + (leg)2Pythagorean Theorem

64 ?16 + 48

64 ?64

YES, these side lengths make a right triangle!

Substitute.

Square

Add.

64 = 64

Distance Formula and Pythagorean Theorem

• Distance Formula:

• Pythagorean formula:

Distance Formula and Pythagorean Theorem

• Distance Formula:

• Pythagorean formula:

We will learn to…

• Apply the converse of the Pythagorean Theorem to classify a triangle by its angles.

Using Pythagorean Converse Activity

• Work with a partner• Each pair should have a bag with pieces of

spaghetti of different lengths– Make sure to measure the pieces to ensure that

you are using the correct pieces for each trial• Fill in the table for each trial• Answer the ‘Analysis’ questions that follow

Is that a triangle??

NO! NO!

YES!

Yes: 7 > 5 9 16 25 25 right

Yes: 9 > 6 16 25 36 41 acute

Yes: 5 > 4 4 9 16 13 obtuse

Yes – For all trials, a + b > c

No, c2 = a2 + b2 is only true for right triangles.

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Used to classify a triangle as right, obtuse or acute

Example 1

• Can segments with lengths of 6.1 feet, 5.2 feet, and 4.3 feet form a triangle? If so, classify the triangle by its angles.

Example 2

• Determine whether segments with lengths 3, 8, and 7 can form a triangle. If they can, classify the triangle by angles.

Example 3

• Classify each triangle below by its angles and sides.

Pythagorean Triples: Any set of _____ ____________ numbers {a, b, c} that satisfies _______________.IF it satisfies the rule stated above, then the three numbers create a _______ ____________.

Example 5: Determine if the following sets of numbers are Pythagorean Triples. Justify your response.

three wholec2 = a2 + b2

right triangle

Example 5: Determine if the following sets of numbers are Pythagorean Triples. Justify your response.

b. {7, 9, 8} c. {37, 12, 35}

c2 ? a2 + b2

92 ? 72 + 82

81 ? 49 + 64

81 < 113

Since c2 < a2 + b2 then this set of numbers is NOT a Pythagorean Triple

c2 ? a2 + b2

372 ? 122 + 352

1369 ? 144 + 1225

1369 = 1369

Since c2 = a2 + b2 then this set of numbers is a Pythagorean Triple