chapter 7 section 7 parallel and perpendicular lines
TRANSCRIPT
Chapter 7 Section 7
Parallel and Perpendicular Lines
What You’ll Learn
You’ll learn to write an equation of a line that is parallel or perpendicular to the graph of a given equation and that passes through a
given point.
Why It’s Important
Surveying
Surveyors use parallel and perpendicular lines to plan construction.
Parallel Lines
Words: If two lines have the same slope, then they are parallel.Model: Symbols: II
y = 2x + 2
y = 2x - 4
Example 1
Determine whether the graph of the equations are parallel. y = -¾x – 24y = -3x + 12
First, determine the slope of the lines. Write each equation in slope intercept form.
The slopes are the same, so the lines are parallel.
y = -¾x -2 Slope intercept form
Slope = -¾
4y = -3x + 12
4y = -3x + 12 4 4
y = -¾x + 3
Slope = -¾
You can always
check by graphing.
Your Turn
Determine whether the graph of the equations are parallel.
y = 2x7 = 2x – y
Yes
Your Turn
Determine whether the graph of the equations are parallel.
y = -3x + 32y = 6x – 5
No
What is a parallelogram???
A parallelogram is a four-sided figure with two sets of parallel sides.
Example 2
Determine whether figure ABCD is a parallelogram. Explore: To be a parallelogram, AB and DC must be parallel. Also, AD and BC must be parallel.
Plan: Find the slope of each segment.
A(-2,2)
B(1,-1)
C(7,2)
D(4,5)
Solve: AB m = -1 – 2 -3 1 – (-2) 3
= = -1
Solve: BC m = -1 – 2 -3 1 - 7 -6
= = ½
Solve: DC m = 2 – 5 -3 7 - 4 3
= = -1
Solve: AD m = 5 - 2 3 4 – (-2) 6
= = ½
Example 2: Continued
AB is parallel to DC because their slopes are both -1. BC is parallel to AD because their slopes are both ½. Therefore, figure ABCD is a parallelogram.
If you know the slope to a line, you can use that information to write an equation of a line that
is parallel to it.
Example 3Write an equation in slope-intercept form of the line that is parallel to the
graph y = -4x + 8 and passes through the point (1, 3).
The slope of the given line is -4. So, the slope of the new line will also be -4. Find the new equation by using the point-slope form.
y – y1 = m(x - x1) Point Slope Formy – 3 = -4(x - 1) Replace (x1, y1) with (1, 3) and m with -4y – 3 = -4x + 4 Distributive Property + 3 + 3 Add 3 to both sidesy = -4x + 7
An equation whose graph is parallel to the graph of 4x + y = 8 and passes through the point at (1, 3) is y = -4x + 7.
Checking: Check by substituting (1, 3) into y = -4x + 7 or by graphing
Your Turn
Write an equation in slope-intercept form of the line that is parallel to the graph of each
equation and passes through the given point.
y = 6x – 4; (2, 3)
y = 6x - 9
Your Turn
Write an equation in slope-intercept form of the line that is parallel to the graph of each
equation and passes through the given point.
3x + 2y = 9; (2, 0)
y = -3/2x + 3
Perpendicular Lines
Words: If the product of the slopes of two lines is -1, then the lines are perpendicular.
Model: Below
y = -x + 3
y = x -1
Example 4Determine whether the graphs of the equations are perpendicular.
y = 2/3x + 1y = -3/2x + 2
The graphs are perpendicularbecause the product of their slope is2 -3 3 2
=∙ -1
y = 2/3x + 1
y = -3/2x + 2
Your TurnDetermine whether the graphs of the equations are perpendicular.
y = 1/5x + 2y = 5x + 1
No
Your TurnDetermine whether the graphs of the equations are perpendicular.
y = -4x + 34y = x -5
Yes
Example 5Write an equation in slope-intercept form of the line that is perpendicular
to the graph of y = ⅓x – 2 and passes through the point (-4, 2).
The slope is ⅓. A line perpendicular to the graph of y = ⅓x – 2 has slope -3. Find the new equation by using the point-slope form.
y – y1 = m(x - x1) Point Slope Form
y – 2 = -3(x – (-4)) Replace (x1, y1) with (-4, 2) and m with -3
y – 2 = -3x - 12 Distributive Property + 2 + 2 Add 2 to both sidesy = -3x – 10The new equation y = -3x -10.
Check by substituting (-4, 2) into the equation or by graphing.
Your TurnWrite an equation in slope-intercept form of the line that is perpendicular
to the graph of each equation and passes through the given point.
y = 2x + 6; (0, 0)
y = -½x
Your TurnWrite an equation in slope-intercept form of the line that is perpendicular
to the graph of each equation and passes through the given point.
2x + 3y = 2; (3, 0)
y = 3/2x – 9/2
Video Examples
• Parallel Lines 1• Parallel Lines 2• Parellal Lines 3• Perpendicular Lines 1• Perpendicular Line 2