Parallel and Perpendicular Lines

Parallel and Perpendicular LinesUsing parallelism and

perpendicularity to solve problems

In the graph below, the two lines are parallel. Parallel lines - are lines in the same plane that never intersect. The equation of line 1 is y = 2x

+ 1. the equation of line 2 is y = 2x -2

Slopes of Parallel Lines

Nonvertical lines are parallel if they have the same slope and

different y-intercepts.Any two vertical lines

are parallel.Any two horizontal lines

are parallel

You can use slope-intercept form of an equation to determine whether lines are

parallel.Are the graphs of y = -1/3x + 5 and 2x + 6y =

12 parallel? Explain.

Write 2x + 6y = 12 in slope-intercept

form, then compare with

y = -1/3x + 5

2x + 6y = 126y = -2x + 126y = - 2x + 12

6 6y = - 1/3x + 2Compare to

y = -1/3x + 5The lines are parallel.

The equations have the same slope, -1/3,

and different y-intercepts.

Are the graphs of -6x + 8y = -24 and y = 3/4x – 7 parallel? Explain.

You can use the fact that the slopes of parallel lines are the same to write the

equation of a line parallel to a given line. To write the equation, you use the slope of the

given line and the point-slope form of a linear equation.

Step 1 Identify the slope of the given line.y = 3/5x – 4

Step 2 Write the equation of the line through (5, 1) using point-slope form.y – y1 = m(x – x1) point-slope form.

y – 1 = 3/5(x – 5) Substitute (5, 1) for (x1,Y1) and 3/5 for m.

y – 1 = 3/5x – 3/5(5) Use the distributive property.

y – 1 = 3/5x – 3 Simplify.

y = 3/5x – 2 Add 1 to each side.

TRY ONE

Write an equation for the line that contains (2, -6) and is parallel to y = 3x + 9

Step 1 Identify the slope of the given line.

Step 2 Write the equation of the line through (2, -6) using point-slope form of a linear equation.

y – y1 = m(x – x1)

Write an equation for the line that is parallel to the given line and that passes

through the given point.

1) Y = 6x - 2; (0, 0)

2) Y = -3x; (3, 0)

3) Y =-2x + 3; (-3, 5)

4) Y = -7/2x + 6; (-4, -6)

The two lines in the graph below are perpendicular. Perpendicular lines – are lines that intersect to form right angles. The line y = 2x + 1 is perpendicular to the line y = -1/2x +

1.

Slopes of perpendicular lines

Two lines are perpendicular if the

product of their slopes is -1. A vertical and a

horizontal line are also perpendicular.

The product of two numbers is -1 if one number is the negative reciprocal of the

other. Here is how to find the negative reciprocal of a number.

Start with a fraction:-1/2

Find its reciprocal:-2/1

Write the negative reciprocal:2/1 or 2

Since -1/2 • 2/1 = -1, 2/1 is the negative reciprocal of -1/2

TRY THESE

Find the negative reciprocal of each:

1) 4

2) 3/4

3) -1/2

4) -2

5) -4/3

You can use the negative reciprocal of the slope of a given line to write an equation of a line perpendicular to that line. To write the equation, you use the negative reciprocal of the slope of

the given line and the point-slope form of a linear equation.

Step 1 Identify the slope of the given line.y = 5x + 3

Step 2 Find the negative reciprocal of the slope.5 • -1/5 = -1

Step 3 Use the point-slope form to write an equation that contains (0, -2) and is perpendicular to y = 5x + 3

y – y1 = m(x – x1) Point-slope form.

y – (-2) = -1/5(x – 0) Substitute (0, -2) for (x1,y1) and -1/5 for m.

y + 2 = -1/5x – 0 Use the distributive property.

y = -1/5x – 2 Subtract 2 from each side. Simplify.

TRY ONE

Write an equation of the line that contains (6, 2) and is perpendicular to y = -2x + 7

Step 1 Identify the slope of the given line.

Step 2 Find the negative reciprocal of the slope.

Step 3 Use the point-slope form of an equation that contains (6, 2) and is perpendicular to y = -2x + 7

Write an equation for the line that is perpendicular to the given line and that

passes through the given point.

1) Y = 2x + 7; (0, 0)

2) Y = -1/3x + 2; (4, 2)

3) Y = x – 3; (4, 6)

4) 4x – 2y = 9; (8, 2)

Write the equation of each line. Determine if the lines are parallel or perpendicular.

Explain why or why not.

Problem Solving

A city’s civil engineer is

planning a new parking garage and a new street. The new street will go from the entrance

of the parking garage to Handel

St. It will be perpendicular to

Handel St. What is the equation of the line representing the new street?

Handel St.

Entrance

Problem Solving