12.1 Finding Square Rootsmathphillips.weebly.com/uploads/1/7/8/2/17823839/ch._12...Section 12.1...

Chapter 12 Radicals and the Pythagorean Theorem

Finding Square Roots12.1

How can you fi nd the side length of a square when you are given the area of the square?

When you multiply a number by itself, you square the number.

42 = 4 ⋅ 4

= 16 4 squared is 16.

To “undo” this, take the square root of the number.

√—

16 = √—

42 = 4 The square root of 16 is 4.

Symbol for squaring is 2nd power.

Symbol for square root is a radical sign.

Work with a partner. Use a square root symbol to write the side length of the square. Then fi nd the square root. Check your answer by multiplying.

a. Sample: s = √—

121 = 11 ft

s

s

Area = 121 ft2

The side length of the square is 11 feet.

b.

s

s

Area = 81 yd2 c.

s

s

Area = 324 cm2 d.

s

s

Area = 361 mi2

e.

s

s

Area = 2.89 in.2 f.

s

s

Area = 4.41 m2 g.

s

s

Area = ft249

ACTIVITY: Finding Square Roots11

Check 11 × 11

11 110

121 !

COMMON CORE STATE STANDARDS

8.EE.2

MSCC7Ape_1201.indd 510MSCC7Ape_1201.indd 510 3/13/12 4:14:31 PM3/13/12 4:14:31 PM

Section 12.1 Finding Square Roots

Work with a partner.

The period of a pendulum is the time (in seconds) it takes the pendulum to swing back and forth.

The period T is represented by T = 1.1 √ —

L , where L is the length of the pendulum (in feet).

Copy and complete the table. Then graph the function. Is the function linear?

L 1.00 1.96 3.24 4.00 4.84 6.25 7.29 7.84 9.00

T

1 2 30 4 5 6 7 8 9 L

T

2

3

4

5

6

7

8

1

0

Length (feet)

Peri

od (s

econ

ds)

Period of a Pendulum

ACTIVITY: The Period of a Pendulum22

3. IN YOUR OWN WORDS How can you fi nd the side length of a square when you are given the area of the square? Give an example. How can you check your answer?

Use what you learned about fi nding square roots to complete Exercises 4 – 6 on page 514.

LL



The Pythagorean Theorem12.2

How are the lengths of the sides of a right triangle related?

Pythagoras was a Greek mathematician and philosopher who discovered one of the most famous rules in mathematics. In mathematics, a rule is called a theorem. So, the rule that Pythagoras discovered is called the Pythagorean Theorem.


a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides (the legs) a and b.

b. Label the length of the longest side (the hypotenuse) c.

c . Draw squares along each of the three sides. Label the areas of the three squares a 2, b 2, and c 2.

d. Cut out the three squares. Make eight copies of the right triangle and cut them out. Arrange the fi gures to form two identical larger squares.

e. What does this tell you about the relationship among a 2, b 2, and c 2?

ACTIVITY: Discovering the Pythagorean Theorem11

Pythagoras(c. 570 B.C.–c. 490 B.C.)

ac

c2

a2

b2

b

c2

b2

a2


8.G.68.G.7


Section 12.2 The Pythagorean Theorem

Work with a partner. Use the result of Activity 1 to fi nd the length of the hypotenuse of each right triangle.

a.

4

c3

b. c

24

10

c.

c

12

23

d.

0.8

c0.6

ACTIVITY: Finding the Length of the Hypotenuse22

Work with a partner. Use the result of Activity 1 to fi nd the length of the leg of each right triangle.

a.

a

16

20

b.

b

0.240.4

ACTIVITY: Finding the Length of a Leg33

Use what you learned about the Pythagorean Theorem to complete Exercises 3 – 5 on page 520.

4. IN YOUR OWN WORDS How are the lengths of the sides of a right triangle related? Give an example using whole numbers.



Approximating Square Roots12.3

How can you fi nd decimal approximations of square roots that are irrational?

You already know that a rational number is a number that can be written as the ratio of two integers. Numbers that cannot be written as the ratio of two integers are called irrational.

Real Numbers

IrrationalRational

Integer

Natural

12

3 −2

2.25

0.3

−3

−1−

−

23

12

π

π−2

3

2


Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer.

a. Archimedes tried to fi nd a rational number whose square is 3. Here are two that he tried.

265 —

153 and 1351

— 780

Are either of these numbers equal to √—

3 ? How can you tell?

b. Use a calculator with a square root key to approximate √

— 3 .

Write the number on a piece of paper. Then enter it into the calculator and square it. Then subtract 3. Do you get 0? Explain.

c. Calculators did not exist in the time of Archimedes. How do you think he might have approximated √

— 3 ?

ACTIVITY: Approximating Square Roots11

Archimedes(c. 287 B.C.–c. 212 B.C.)

SquareRoot Key


8.EE.28.NS.18.NS.2


Section 12.3 Approximating Square Roots


a. Use grid paper and the given scale to draw a horizontal line segment 1 unit in length. Label this segment AC.

b. Draw a vertical line segment 2 units in length. Label this segment DC.

c. Set the point of a compass on A. Set the compass to 2 units. Swing the compass to intersect segment DC. Label this intersection as B.

d. Use the Pythagorean Theorem to show that the length of segment BC is √

— 3 units.

e. Use the grid paper to approximate √—

3 .

AC

D

B

AA

Scale:

of a unit110

22

1

ACTIVITY: Approximating Square Roots Geometrically22

Use what you learned about approximating square roots to complete Exercises 5 – 8 on page 529.

3. Repeat Activity 2 for a triangle in which segment CA is 2 units and segment BA is 3 units. Use the Pythagorean Theorem to show that segment BC is √

— 5 units. Use the grid paper to approximate √

— 5 .

4. IN YOUR OWN WORDS How can you fi nd decimal approximations of square roots that are irrational?



Simplifying Square Roots12.4

How can you use a square root to describe the golden ratio?

Two quantities are in the golden ratio if the ratio

x + 1

x 1

between the sum of the quantities and the greater quantity is the same as the ratio between the greater quantity and the lesser quantity.

x + 1 —

x = x —

1

In a future algebra course, you will be able to prove that the golden ratio is

1 + √—

5 —

2 Golden ratio.


a. Use grid paper and the given scale to draw a square that is 1 unit by 1 unit (blue).

b. Draw a line from midpoint C of one side of the square to the opposite corner D, as shown.

c. Use the Pythagorean Theorem to fi nd the length of segment CD.

d. Set the point of a compass on C. Set the compass radius to the length of segment CD. Swing the compass to intersect line BC at point E.

e. The rectangle ABEF is called a golden rectangle because the ratio of its side lengths is the golden ratio.

f. Use a calculator to fi nd a decimal approximation of the golden ratio. Round your answer to two decimal places.

ACTIVITY: Constructing a Golden Ratio11

1

A

C

E

D

F

B

D

CC

12

52

Scale:

of a unit110

1


8.NS.2


Section 12.4 Simplifying Square Roots

Use what you learned about square roots to complete Exercises 3 –5 on page 536.

3. IN YOUR OWN WORDS How can you use a square root to describe the golden ratio? Use the Internet or some other reference to fi nd examples of the golden ratio in art and architecture.


Leonardo da Vinci was one of the fi rst to notice that there are several ratios in the human body that approximate the golden ratio.

a. Use a tape measure or two yardsticks to measure the lengths shown in the diagram for both you and your partner. (Take your shoes off before measuring.)

b. Copy the tables below. Record your results in the fi rst two columns.

c. Calculate the ratios shown in the tables.

d. Leonardo da Vinci stated that for many people, the ratios are close to the golden ratio. How close are your ratios?

You

a = b = a — b

=

c = d = c — d

=

e = f = e — f =

g = h = g — h

=

Partner

a = b = a — b

=

c = d = c — d

=

e = f = e — f =

g = h = g — h

=

ACTIVITY: The Golden Ratio and the Human Body22

b

a

c

d

e

f

g

h

Navel



Using the Pythagorean Theorem12.5

How can you use the Pythagorean Theorem to solve real-life problems?


a. A baseball player throws a ball from second base to home plate. How far does the player throw the ball? Include a diagram showing how you got your answer. Decide how many decimal points of accuracy are reasonable. Explain your reasoning.

b. The distance from the pitcher’s mound to home plate is 60.5 feet. Does this form a right triangle with fi rst base? Explain your reasoning.

ACTIVITY: Using the Pythagorean Theorem11


The recommended angle for a fi refi ghting ladder is 75°.

When a 110-foot ladder is put up against a building at this angle, the base of the ladder is about 28 feet from the building.

The base of the ladder is 8 feet above the ground.

How high on the building will the ladder reach? Round your answer to the nearest tenth.

ACTIVITY: Firefi ghting and Ladders22

90 ft

90 ft

110 ft x

8 ft

28 ft


8.G.68.G.78.G.8


Section 12.5 Using the Pythagorean Theorem

s

s

s


Find the perimeter of each fi gure. Round your answer to the nearest tenth. Did you use the Pythagorean Theorem? If so, explain.

a. Right triangle b. Trapezoid c. Parallelogram

ACTIVITY: Finding Perimeters33

Use what you learned about using the Pythagorean Theorem to complete Exercises 3 –5 on page 542.

5. IN YOUR OWN WORDS How can you use the Pythagorean Theorem to solve real-life problems?

6. Describe a situation in which you could use the Pythagorean Theorem to help make decisions. Give an example of a real-life problem.


a. Write a formula for the area of an equilateral triangle with side length s.

b. Use your formula to fi nd the area of an equilateral triangle with a side length of 10 inches.

ACTIVITY: Writing a Formula44

3 cm

3 cm

2 cm

6 ft

4 ft

4 in.

2 in.

3 in.


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