5.5 parallel perp lines

–10 –705

20

–3–15

x y

Point-Slope Form and Writing Linear Equations

LESSON 5-4 Lesson Quiz

23

25yes; y + 3 = (x – 0)

65

65

75

y + 5 = – (x – 3); y = – x –

y – 4 = – (x – 0), or y = – x + 423

23

1. Graph the equation y + 1 = –(x – 3).

2. Write an equation of the line with

slope – that passes through

the point (0, 4).

3. Write an equation for the line that passes through (3, –5) and (–2, 1) in point-slope form and slope-intercept form.

4. Is the relationship shown by the data linear? If so, model that data with an equation.

5-5Parallel and

Perpendicular Lines

California Content Standards7.0 Derive linear equations by using the point-slope formula. Master8.0 Understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Find the equation of a line perpendicular to a given line that passes through a given point. Develop Master

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What You’ll Learn…And Why

What You’ll Learn…And Why

To determine whether lines are parallelTo determine whether lines are

perpendicular

To use parallel and perpendicular lines to plan a bike path.

Slopes of Parallel Lines

Nonvertical lines are parallel if

Any two vertical lines are parallel.

Example the equations

Have the same slope, , and different y-intercept. The graphs of the two equations are parallel.

They have the same slope and different y-intercepts

2

3

y =23x+1 and y=

23x−3

Slopes of Perpendicular Lines

Two lines are perpendicular if the product of their slopes is

A vertical and horizontal line are also perpendicular.

Example The slope of

The slope of y = 4x + 2 is 4. Since

The graphs of the two equations are perpendicular.

- 1

y =−14x−1 is −

14.

−1

4• 4 = −1

VocabularyParallel lines are lines in the same plane that

never intersect. Equations have the same slope, different y-intercepts.

Perpendicular lines are lines that intersect to form right angles. Equations have the opposite and reciprocal slopes.

The product of a number and its negative reciprocal is -1.

Write an equation for the line that contains (–2, 3) and is parallel to y = x – 4.

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52

Step 1  Identify the slope of the given line.

y = x – 4

slope

Step 2 Write the equation of the line through (–2, 3) using slope-intercept form.

Write an equation for the line that contains

(–2, 3) and is parallel to y = x – 4.

y – 3 = x + 5 Simplify.52

y = x + 8 Add 3 to each side and simplify.52

y – 3 = (x + 2) Substitute (2,3) for (x1, y1) and

for m.

52

52

52

(Continued)

y – y1 = m(x – x1) Use point-slope form.

The line in the graph represents the street in front of a new house. The point is the front door. The sidewalk from the front door will be perpendicular to the street. Write an equation representing the sidewalk.

Step 1  Find the slope m of the street.

m = = = = – Points (0, 2) and (3, 0) are on the street.

y2 – y1

x2 – x1

0 – 23 – 0

–2 3

23

The equation for the sidewalk is y = x – 3.32

Step 2 Find the negative reciprocal of the slope.

The negative reciprocal of – is . So the slope of the

sidewalk is . The y-intercept is –3.

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32

32

CA Standards Check1) Write an equation for the line that contains (2,-6) and

is parallel to y=3x +9.

Parallel lines: same slope, different y-intercepts.

PointSlope

y – y1 = m (x – x1 ) - 6 2 3

y + 6 = 3(x – 2 )

y + 6 = 3x – 6 – 6 – 6

y = 3x – 12

CA Standards Check2a) Write an equation for the line that contains (1,8)

and is perpendicular to y=3/4x + 1.

Perpendicular lines: Opposite and reciprecal slopes.

PointSlope

y – y1 = m (x – x1 ) 8 1 −4

3

y – 8 = (x – 1 ) −4

3

y – 8 = x + −4

34

3

y = x + −4

3

28

3

8 =3

24

+ 8 + .24

3

y =−43x+ 9

13

CA Standards Check2b) Using the diagram in Ex 2, rite an equation in slope-intercept

form for a new sidewalk perpendicular to the street from a front door at (-1, -2).

Point

Slope y – y1 = m (x – x1 ) (-1) (-2) −2

3

y + 1 = (x + 2 ) −2

3 -1 =3

-3

y = x - 3 3

2

y + 1 = x - −2

34

3

- 1 - . 3

3

y = x - −2

3

8

3 y =−23x−2

23

Closure:

Compare the equations of non-vertical parallel line, and perpendicular lines.

Parallel lines have the same slope, but different y-intercepts.Perpendicular lines have slopes that are negative reciprocals of each other.