CorrectionKey=TX-B Name Class Date 4 . 5 Equations … Parallel and...B Graph the equations y = 2 (x + 1) and y = 2x-3. C What do you notice about the graphs of the two lines? About

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G.2.C Determine an equation of a line parallel or perpendicular to a given line that passes through a given point. Also G.2.B

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Explore Exploring Slopes of Lines

The slope of a straight line in a coordinate plane is the ratio of the rise to the run.

To find a numeric expression for slope, take two arbitrary points on the line. The coordinates of the first point can be represented by (x 1 , y 1). The coordinates of the second point can be represented by ( x 2 , y 2 ) .

The rise is the vertical change from point A to point B, and can be expressed as the difference __________.

The run is the horizontal change from point A to point B, and can be expressed as the difference __________.

So the slope m is equal to ______________. This is the slope formula.

Module 4 233 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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B Graph the equations y = 2 (x + 1) and y = 2x - 3.

C 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.

D Use a protractor to measure the angle formed by the intersection of the lines. What kind of angle is it?

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

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

G 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 .

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.


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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.

B 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 .

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 =

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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 .

B 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 .

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?



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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.

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?

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice



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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 ) 8. y = 2 - 2x _ 3 ; (-6, 1)

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

9. y = 10x; (1, -3) 10. y = 2.5x + 11; (-20, 7)

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

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



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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.

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

18. 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.



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19. 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)?

20. Show that when deriving the slope formula, it does not matter in which order you take the two points.

21. 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

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yH.O.T. Focus on Higher Order Thinking

22. 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?

23. 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.

24. 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.



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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 these points on a coordinate grid of the park in units of 1 rod: Park Headquarters (15, 0) and Park Entrance (25, 25) . The Interpretive Center is located on the y-axis, and 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



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CorrectionKey=TX-B Name Class Date 4 . 5 Equations … Parallel and...B Graph the equations y = 2 (x + 1) and y = 2x-3. C What do you notice about the graphs of the two lines? About