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Explore Exploring Slopes of LinesRecall that the slope of a straight line in a coordinate plane is the ratio of the rise to the run. In the figure, the slope of _ AB is rise ___ run = 4 __ 8 = 1 __ 2 .

A Graph the equations y = 2 (x + 1) and y = 2x - 3.

B What do you notice about the graphs of the two lines? About the slopes of the lines?

C The graphs of x + 3y = 22 and y = 3x - 14 are shown. Use a protractor. What is the measure of the angle formed by the intersection of the lines. What does that tell you about the lines?

D What are the slopes of the two lines? How are they related?

E Complete the statements: If two nonvertical lines

are , then they have equal slopes. If two nonvertical lines are perpendicular,

then the product of their slopes is .

Resource Locker

y

0-3 -1-5

3

1

-5

3

x

B(4, 1)

A(-4, -3)

Run = 4-(-4) = 8

Rise = 1-(-3) = 4

y

0-2-4

2

4

-4

-2

42

x

y

0 2 4 6 8

2

4

6

8

x

The lines are parallel. The slopes are equal.

90°; the lines are perpendicular.

- 1 __ 3

and 3; the slopes are opposite reciprocals.

parallel

–1

Module 4 205 Lesson 5

4.5 Equations of Parallel and Perpendicular Lines

Essential Question: How can you find the equation of a line that is parallel or perpendicular to a given line?

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GE_MNLESE385795_U2M04L5.indd 205 2/25/16 11:37 PM

Common Core Math StandardsThe student is expected to:

G-GPE.B.5

… find the equation of a line parallel or perpendicular to a given line that passes through a given point … .

Mathematical Practices

MP.2 Reasoning

Language ObjectiveExplain to a partner how to use the slope of a line to find the equation of a parallel or perpendicular line.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 179184

Turn to these pages to find this lesson in the hardcover student edition.

Equations of Parallel and Perpendicular Lines

ENGAGE Essential Question: How can you find the equation of a line that is parallel or perpendicular to a given line?Possible answer: The slopes of parallel lines are

equal. Substitute the known slope and the

coordinates of a point on the other line into the

point-slope form to find the equation of the parallel

line. The product of the slopes of perpendicular

lines is -1. Substitute the opposite reciprocal of the

known slope and the coordinates of a point on the

other line into the point-slope form to find the

equation of the perpendicular line.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo. Explain that GPS stands for Global Positioning System, a system of 24 orbiting satellites that enables a person to pinpoint his or her precise location on Earth’s surface. Then preview the Lesson Performance Task.

205

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Name

Class Date

Explore Exploring Slopes of Lines

Recall that the slope of a straight line in a coordinate plane is

the ratio of the rise to the run. In the figure, the slope of

_ AB is rise ___ run = 4 __ 8 = 1 __ 2 .

Graph the equations y = 2 (x + 1) and y = 2x - 3.

What do you notice about the graphs of the two lines?

About the slopes of the lines?

The graphs of x + 3y = 22 and y = 3x - 14 are shown.

Use a protractor. What is the measure of the angle formed

by the intersection of the lines. What does that tell you

about the lines?

What are the slopes of the two lines? How are they related?

Complete the statements: If two nonvertical lines

are , then they have equal slopes. If two nonvertical lines are perpendicular,

then the product of their slopes is .

Resource

Locker

G-GPE.B.5 … find the equation of a line parallel or perpendicular to a given line that passes

through a given point … .COMMONCORE

y

0-3 -1

-5

3

1

-5

3

xB(4, 1)

A(-4, -3)

Run = 4-(-4) = 8

Rise = 1-(-3)

= 4

y

0-2-4

2

4

-4

-2

42

x

y

0 2 4 6 8

2

4

6

8

x

The lines are parallel. The slopes are equal.

90°; the lines are perpendicular.

- 1 __ 3 and 3; the slopes are opposite reciprocals.

parallel–1

Module 4

205

Lesson 5

4 . 5 Equations of Parallel

and Perpendicular Lines

Essential Question: How can you find the equation of a line that is parallel

or perpendicular to a given line?

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205 Lesson 4 . 5

L E S S O N 4 . 5

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Reflect

1. Your friend says that if two lines have opposite slopes, they are perpendicular. He uses the slopes 1 and –1 as examples. Do you agree with your friend? Explain.

2. The frets on a guitar are all perpendicular to one of the strings. Explain why the frets must be parallel to each other.

Explain 1 Writing Equations of Parallel LinesYou can use slope relationships to write an equation of a line parallel to a given line.

Example 1 Write the equation of each line in slope-intercept form.

The line parallel to y = 5x + 1 that passes through (-1, 2)

Parallel lines have equal slopes. So the slope of the required line is 5.

Use point-slope form. y - y 1 = m (x - x 1 )

Substitute for m, x 1 , y 1 . y - 2 = 5 (x - (-1) ) Simplify. y - 2 = 5x + 5

Solve for y. y = 5x + 7

The equation of the line is y = 5x + 7.

The line parallel to y = -3x + 4 that passes through (9, -6)

Parallel lines have slopes. So the slope of the required line is .

Use point-slope form. y - y 1 = m(x - x 1 )

Substitute for m, x 1 , y 1 . y - = ( x - ) Simplify. y + 6 = x +

Solve for y. y = x +

The equation of the line is .

No; although lines with slopes of 1 and -1 are perpendicular, it’s because the product

of the slopes is -1. Slopes of 2 and -2 are opposites, but the corresponding lines are not

perpendicular.

The frets are lines that are perpendicular to the same line (the string), so the frets must

be parallel to each other.

equal -3

-6 -3 9

27

21

-3

-3

y = -3x + 21


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Math BackgroundIn this lesson, students use the slope criterion for parallel lines and the slope criterion for perpendicular lines to solve problems. Note that the slope criteria given here assume that the lines are neither vertical nor horizontal. If the lines are vertical, the criteria for parallel and perpendicular lines do not apply, since slope is not defined for vertical lines. If the lines are horizontal, both lines have a slope of zero, and the criterion for parallel lines is trivial.

EXPLORE Exploring Slopes of Lines

INTEGRATE TECHNOLOGYStudents have used geometry software to construct perpendicular lines and calculate their slopes. They can use the calculation feature to find the product of slopes of perpendicular lines is always -1.

QUESTIONING STRATEGIESWhat appears to be true about the slopes of non-vertical parallel lines? They are equal.

What appears to be true about the slopes of two non-vertical perpendicular lines? The

slopes are opposite reciprocals.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 You may want to discuss the biconditional nature of the slope criteria. Because they are if and only if statements, the criteria can be used in either direction. That is, if you know that two lines are parallel (perpendicular), you can conclude that they have the same (opposite reciprocal) slope. Conversely, if you know that two lines have the same (opposite reciprocal) slope, you can conclude that they are parallel (perpendicular).

EXPLAIN 1 Writing Equations of Parallel Lines

QUESTIONING STRATEGIESHow do you know if an equation is written in slope-intercept form? It is of the form

y = mx + b, with m the slope and b the y-intercept.

How can you use graphing to check your answer? Graph the given line and your

answer line. They should be parallel.

PROFESSIONAL DEVELOPMENT

Equations of Parallel and Perpendicular Lines 206


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Reflect

3. What is the equation of the line through a given point and parallel to the x-axis? Why?

Your Turn

Write the equation of each line in slope-intercept form.

4. The line parallel to y = -x that passes through (5, 2.5)

5. The line parallel to y =

3

__

2

x + 4 tha t passes through

(-4, 0)

Explain 2 Writing Equations of Perpendicular LinesYou can use slope relationships to write an equation of a line perpendicular to a given line.

Example 2 Write the equation of each line in slope-intercept form.

The line perpendicular to y = 4x - 2 that passes through (3, -1)

Perpendicular lines have slopes that are opposite reciprocals, which means that the product of the slopes will be -1. So the slope of the required line is - 1 __ 4 .

y - y 1 = m (x - x 1 ) Use point-slope form.

y - (-1) = - 1 _ 4 (x - 3) Substitute for m, x 1 , y 1 .

y + 1 = - 1 _ 4 x + 3 _ 4 Simplify.

y = - 1 _ 4 x - 1 _ 4 Solve for y.

The equation of the line is y = - 1 __ 4 x - 1 __ 4 .

The line perpendicular to y = - 2 __ 5 x + 12 that passes through (-6, -8)

The product of the slopes of perpendicular lines is . So the slope of the required line is .

y - y 1 = m (x - x 1 ) Use point-slope form.

y - = ( x - ) Substitute for m, x 1 , y 1 .

y + 8 = x + Simplify.

y = x + Solve for y.

The equation of the line is y .

The equation is y = y 1 , where y 1 is the y-coordinate of the given point. This is because the

x-axis is a horizontal line with equation y = 0.

y - 2.5 = -1 (x - 5)

y - 2.5 = -x + 5

y = -x + 7.5

y - (0) = 3 _ 2

(x - (-4) )

y = 3 _ 2

x + 6

-1

-8 -6

15

7

5 __ 2

5 __ 2

5 __ 2

y = 5 __ 2 x + 7

5 __ 2




Lines are parallel.

Lines are perpendicular.

Lines have the same slope.

The product of slopes is –1.

COLLABORATIVE LEARNING

Whole Class ActivityHave groups of students create posters to describe the slope criteria. Then remind students about the biconditional nature of the criteria and ask them if the criteria are true in two directions. Ask them to display the criteria as graphic organizers. Sample organizers:

AVOID COMMON ERRORSRemind students that the x-coefficient gives the slope of a line only when the equation of the line is written in slope-intercept form. For example, some students might say that the slope of the line represented by the equation y - 2x = 4 is -2. However, the equation is not in slope-intercept form. Rewriting the equation in this form gives y = 2x + 4, which shows that the correct slope is 2.

EXPLAIN 2 Writing Equations of Perpendicular Lines

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Students can use their graphing calculators to check that two equations represent perpendicular lines. However, students should be aware that perpendicular lines may or may not appear to be perpendicular on a graphing calculator, depending upon the viewing window that is used. To ensure that perpendicular lines appear to be perpendicular, students should go to the ZOOM menu and choose 5:ZSquare.

QUESTIONING STRATEGIESThe given line has a positive slope. What does this tell you about the required perpendicular

line? Why? It must have a negative slope because

the product of the slopes is −1.

How can you check your answers? Check that

the product of the slopes is −1.

207 Lesson 4 . 5


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Reflect

6. A carpenter’s square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Why must the two lines be parallel?

Your Turn

Write the equation of each line in slope-intercept form.

7. The line perpendicular to y = 3 __ 2 x + 2 that passes through (3, –1)

8. The line perpendicular to y = -4x that passes through (0, 0)

Elaborate

9. Discussion Would it make sense to find the equation of a line parallel to a given line, and through a point on the given line? Explain.

10. Would it make sense to find the equation of a line perpendicular to a given line, and through a point on the given line? Explain.

11. Essential Question Check-In How are the slopes of parallel lines and perpendicular lines related? Assume the lines are not vertical.

Both lines are perpendicular to the edge of the

board. If two coplanar lines are perpendicular to

the same line, then the two lines are parallel to each

other, so the lines must be parallel to each other.

y − (-1) = - 2 __ 3 (x - 3)

y = - 2 __ 3 x + 1

y - 0 = 1 __ 4 (x - 0)

y = 1 __ 4 x

No; if the point is on the line, the line can’t be parallel to the line, because it either

intersects it or it is the same line.

Yes; the line will be perpendicular to the given line at the point.

Parallel lines have the same slope; perpendicular lines have slopes whose product is -1.


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DIFFERENTIATE INSTRUCTION

Communicating MathGroup students in pairs and give each pair a sheet of graph paper with a non-vertical, non-horizontal line drawn on it. Have students draw axes and find the equation of the line. Then have each student plot a point that is not on the line, and find the equation of the line that is parallel, and the equation of the line that is perpendicular to the original line and that passes through the partner’s point. When they are done, they should compare the slopes of their lines to show that the two new lines are parallel (or perpendicular) to each other.

ELABORATE QUESTIONING STRATEGIES

What is the equation of the line through a given point and parallel to the x-axis?

Why? The equation is y = b, where b is the

y-coordinate of the point.

Can either of the lines referred to in the slope criterion for perpendicular lines be vertical?

Why or why not? No; the slope criterion specifies

that neither line is vertical. However, since the lines

are perpendicular, if one line were horizontal, the

other would be vertical.

SUMMARIZE THE LESSONGiven the equation of a line and a point not on the line, how do you find the equation of a

line parallel and perpendicular to the given line? Sample answer: Parallel: Use the

slope-intercept form of a line, y = mx + b, and

replace m with the slope from the given line. Use

the given point and the slope to solve for b and then

rewrite the equation using the same slope m and

the new y-intercept b.

Perpendicular: Do the same steps as for parallel

except replace m with the opposite reciprocal of m.



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• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

Use the graph for Exercises 1–4.

1. A line with a positive slope is parallel to one of the lines shown. What is its slope?

2. A line with a negative slope is perpendicular to one of the lines shown. What is its slope?

3. A line with a positive slope is perpendicular to one of the lines shown. What is its slope?

4. A line with a negative slope is parallel to one of the lines shown. What is its slope?

Find the equation of the line that is parallel to the given line and passes through the given point.

5. y = –3x + 1; (9, 0) 6. y = 0.6x – 3; (–2, 2) 7. y = 5 (x + 1) ; ( 1 _ 2 , - 1 _ 2 )

Find the equation of the line that is perpendicular to the given line and passes through the given point.

8. y = 10x; (1, -3) 9. y = - 1 _ 3 x - 5; (12, 0) 10. y = 5x + 1 _ 3 ; (1, 1)

y

0

2

4

6

642

x

The line will be perpendicular to the line with

slope 5-1 _ 1-2

= -4. So the slope is 1 _ 4

.

The line will be parallel to the line with

slope 5-1 _ 1-2

= -4. So the slope is −4.

6-2 _ 5-0

= 4 _ 5

- 5 _ 4

y − 0 = −3 (x − 9)

y = −3x + 27

y − 2 = 0.6 (x − (−2) )

y = 0.6x + 3.2

y = 5x + 5

y − (− 1__2)= 5 (x − 1__

2) y = 5x − 3

y - (-3) = -0.10 (x - 1)

y = -0.1x - 2.9

y = 3 (x - 12)

y = 3x - 36

3y = 5x + 1

y = 5 __ 3 x + 1 __ 3

y - 1 = - 3 __ 5 (x - 1)

y = -3x + 8 _______ 5


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GE_MNLESE385795_U2M04L5 209 5/22/14 12:20 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–4 1 Recall of Information MP.6 Precision

5–10 1 Recall of Information MP.2 Reasoning

11 2 Skills/Concepts MP.2 Reasoning

12–15 2 Skills/Concepts MP.4 Modeling

16 2 Skills/Concepts MP.4 Modeling

17–19 2 Skills/Concepts MP.2 Reasoning

EVALUATE

ASSIGNMENT GUIDE

Concept & Skills Practice

ExploreExploring Slopes of Lines

Exercises 1–4

Example 1Writing Equations of Parallel Lines

Exercises 5–7, 11–12, 14–15

Example 2Writing Equations of Perpendicular Lines

Exercises 8–10, 13, 16–19

AVOID COMMON ERRORSA common error students make when finding slopes of perpendicular lines is using the same sign for both slopes. One slope must be the opposite reciprocal of the other, not just the reciprocal of the other, so that the product is -1, not 1.

209 Lesson 4 . 5


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11. Determine whether the lines are parallel. Use slope to explain your answer.

The endpoints of a side of rectangle ABCD in the coordinate plane are at A (1, 5) and B (3, 1) . Find the equation of the line that contains the given segment.

12. _ AB 13.

_ BC

14. _ AD 15.

_ CD if point C is at (7, 3)

16. A well is to be dug at the location shown in the diagram. Use the diagram for parts (a–c).

a. Find the equation that represents the road.

b. A path is to be made from the road to the well. Describe how this should be done to minimize the length of the path.

c. Find the equation of the line that contains the path.

y

0-2-4

2

4

-4

-24

x

yWell

Road

-6 -4 -2 2-2

-4

2

4

x0

The top line passes through (-4, 0) and (0, 3) , so its

slope is 3 __ 4 . The bottom line passes through (0, -2)

and (3, 0) , so its slope is 2 __ 3 . The lines do not have the

same slope, so they are not parallel.

The slope of the required line is -2. y - 1 = -2 (x - 3) ; y - 1 = -2x + 6; y = -2x + 7

_ BC ⟘ _ AB , so the slope of the required line

is 1 _ 2

. y - 1 = 1 _ 2

(x - 3) ; y - 1 = 1 _ 2

x - 3 _ 2

;

y = 1 _ 2

x - 1 _ 2

_ CD ‖

_ AB , so the slope of the required

line is -2. y - 3 = -2 (x - 7) ; y = -2x + 17 _ AD ‖

_ BC , so the slope of the required line is 1 _

2 .

y - 5 = 1 _ 2

(x - 1) ; y = 1 _ 2

x + 9 _ 2

y - (-7) = -4 (x-5) ; y = -4x - 23

The line containing the path should be perpendicular to the road.

The slope is

-6 - (-5)

__ -4 - 0 = 1 _

4 and the

y-intercept is - 5.

y = 1 _ 4

x - 5


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GE_MNLESE385795_U2M04L5 210 5/22/14 12:20 PMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

20 3 Strategic Thinking MP.2 Reasoning

21 2 Skills/Concepts MP.3 Logic

22 3 Strategic Thinking MP.2 Reasoning

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Ask students to think about how they can use slope to solve geometry problems. Remind them that since parallel lines have the same slope, they can analyze lines containing sides of polygons, for example, to see whether the sides are parallel. They can also use slope to write the equation of a line that is parallel or perpendicular to a given line. Point out that since perpendicular lines have opposite reciprocal slopes, they can analyze the lines containing the sides of polygons, for example, to see if the polygon contains any right angles.

INTEGRATE MATHEMATICAL PRACTICESFocus on TechnologyMP.5 Ask each student to use geometry software to draw a simple sketch that involves perpendicular lines and right angles. Have students exchange sketches, measure the slopes of the lines, and find the product of the slopes. Have them do another sketch that involves parallel lines. Have students exchange sketches and measure the slopes of the lines.

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Instruct students to write two equations in the form y = mx + b, one with a positive value of m and one with a negative value of m. Have them graph each line on a coordinate plane and then plot a point that is not on either line. For each of the original lines, instruct students to explain how to find the equation of a parallel line through the point and the perpendicular line through the point.



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17. Use the graph for parts (a–c),

a. Find the equation of the perpendicular bisector of the segment. Explain your method.

b. Find the equation of the line that is parallel to the segment, but has the same y-intercept as the equation you found in part a.

c. What is the relationship between the two lines you found in parts (a) and (b)?

18. Line m is perpendicular to x - 3y = -1 and passes through (1, 5) . What is the slope of line m?

A. -3 B. 1 _ 3 C. 3 D. 5

19. Determine whether each pair of lines are parallel, perpendicular, or neither. Select the correct answer for each lettered part.

a. x - 2y = 12; y = x + 5 Parallel Perpendicular Neither

b. 1 _ 5

x + y = 8; y = 5x Parallel Perpendicular Neither

c. 3x - 2y = 12; 3y = -2x + 5 Parallel Perpendicular Neither

d. y = 3x - 1; 15x - 5y = 10 Parallel Perpendicular Neither

e. 7y = 4x + 1; 14x + 8y = 10 Parallel Perpendicular Neither

H.O.T. Focus on Higher Order Thinking

20. Communicate Mathematical Ideas Two lines in the coordinate plane have opposite slopes, are parallel, and the sum of their y-intercepts is 10. If one of the lines passes through (5, 4) , what are the equations of the lines?

21. Explain the Error Alan says that two lines in the coordinate plane are perpendicular if and only if the slopes of the lines are m and 1 __ m . Identify and correct two errors in Alan’s statement.

22. Analyze Relationships Two perpendicular lines have opposite y-intercepts. The equation of one of these lines is y = mx + b. Express the x-coordinate of the intersection point of the lines in terms of m and b.

0

30

30 60 90 120 150

60

90

120

150

x

y

m = 150 - 60 _ 30 - 120

= -1; midpt M = ( 30 + 120

_ 2

, 150 + 60

_ 2

) = (75, 105) ;

perpendicular bisector: y - 105 = 1 (x - 75) or y = x + 30

slope = -1: y = -x + 30

They are perpendicular.

A; the slope of the given line is 1 _ 3 and its opposite reciprocal is -3.

y = 4 and y = 6; parallel lines have equal slopes if the slopes are opposites, they must be zero; a line with slope 0 through (5, 4) has equation y = 4.

He should have said “two nonvertical lines” because vertical lines have undefined slope. He should have had a negative sign on one of his expressions for slope because the slopes of nonvertical perpendicular lines have a product of -1.

If one equation is y = mx + b, then the other is y = - 1 __ m x - b. mx + b = - 1 _ m x - b

mx + 1 _ m x = -2b; x ( m 2 + 1) = -2mb; x = -2mb _ m 2 + 1


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GE_MNLESE385795_U2M04L5.indd 211 2/26/16 3:02 AM

COOPERATIVE LEARNINGHave students work in groups of three or four. Ask them to choose one student to give directions. Instruct the other students to each draw a line and a point that is not on the line. The first student will then give a set of directions, step by step, for finding the equation of a parallel or of a perpendicular to the line through the point. The other students will not know which type of line it is until they have followed the directions. Once the student has successfully guided the other students through the process to find the equation of the line, have another student take the lead and describe the process to the others.

JOURNALHave students write and solve a problem involving finding the equation of a line that is parallel to a given line. Remind students to show all the steps of the solution and to explain how they can check the answer.

211 Lesson 4 . 5


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Surveyors typically use a unit of measure called a rod, which equals 16 1 __ 2 feet. (A rod may seem like an odd unit, but it’s very useful for measuring sections of land, because an acre equals exactly 160 square rods.) A surveyor was called upon to find the distance between a new interpretive center at a park and the park entrance. The surveyor plotted the points shown on a coordinate grid of the park in units of 1 rod. The line between the Interpretive Center and Park Headquarters forms a right angle with the line connecting the Park Headquarters and Park Entrance.

What is the distance, in feet, between the Interpretive Center and the park entrance? Explain the process you used to find the answer.

Lesson Performance Task

N

EW

S

InterpretiveCenter

ParkHeadquarters

(15, 0)

Park Entrance(25, 25)

About 518 feet; one method is shown.

Find the slope of the line between Park Headquarters (PH) and the park entrance (PE):

m = 25 - 0 _ 25 - 15

= 25 _ 10

= 5 _ 2

Since the line connecting the Interpretive Center (IC) and Park Headquarters (PH)

forms a right angle with the line connecting PH and PE, its slope must be the opposite

reciprocal, - 2 __ 5 .

Use the slope and the coordinates of PH (15, 0) to find the coordinates of IC:

m = 0 - y

______ 15 - x

- 2 _ 5 = -y

_____ 15 - 0 = -y

__ 15

-2 = -y

__ 3

-6 = -y

y = 6

So the coordinates of IC are (0, 6).

Use the distance formula to find the distance in rods between IC and PE:

d = √ ――――――― (25 - 0) 2 + (25 - 6) 2 ≈ 31.4

So the distance in feet equals 31.4 × 16.5, or approximately 518 feet.




EXTENSION ACTIVITY

The Lesson Performance Task mentions that an acre equals 160 square rods. Have students show that that is true, using the fact that 1 square mile equals 640 acres.

Possible explanation: 1 mi = 5280 ft so 1 m i 2 = 5280 × 5280 = 27, 878, 400 f t 2

1 acre = 27, 878, 400 f t 2 ÷ 640 = 43, 560 f t 2

1 rod = 16.5 ft, so 1 ro d 2 = 16.5 × 16.5 = 272.25 f t 2

1 acre = (43, 560 ÷ 272.25) = 160 ro d 2

AVOID COMMON ERRORSIn calculating the distance between the park entrance and the Interpretive Center, students may apply the Distance Formula incorrectly as:

d = √ ―――――――― (25 - 0) 2 + (25 - 6) 2

= √ ―――― 2 5 2 + 1 9 2

= √ ―― 2 5 2 + √ ―― 1 9 2

= 25 + 19

= 44

Remind students to find the sum beneath the radical sign first before finding the square root:

d = √ ―――――――― (25 - 0) 2 + (25 - 6) 2

= √ ―――― 2 5 2 + 1 9 2

= √ ―――― 625 + 361

= √ ―― 986

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 The distance between Park Headquarters and the Interpretive Center is approximately 267 feet. Explain how you could find the distance from the park entrance to Park Headquarters without using the Distance Formula. You know the distance from

the Interpretive Center to Park Headquarters (267

feet) and the distance from the Interpretive Center

to the park entrance (518 feet), the lengths of two

sides of a right triangle. You can use the

Pythagorean Theorem to find the length of the third

side joining the park entrance and Park

Headquarters.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



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