Download 3.6C Parallel and Perpendicular Lines in the Coordinate Plane

3-6C Lines in the Coordinate Plane
Objectives: G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems.
For the Board: You will be able to graph lines and write their equations in slope-intercept and point-slope
form. You will be able to classify lines as parallel, intersecting, or coinciding.
A system of two linear equations in two variables represents two lines.
The lines can be parallel, intersecting, or coinciding.
Perpendicular lines are a special form of intersecting lines.
Coinciding lines are the same line but written in a different form.
Pairs of Lines
Parallel Lines
Same slope
Different y-intercept
y = 5x + 8
y = 5x – 4
Intersecting Lines
Different slopes
y = 2x – 5
y = 4x + 3
Perpendicular Lines
Slopes which are opposites
and reciprocals
y = 2x – 5
y = -1/2 x + 3
Coinciding Lines
Same slope
Same y-intercept
y = 2x – 4
y = 2x - 4
Steps: 1. Rearrange both equations into slope-intercept form by solving the equation for y.
2. Compare the slopes using the above table.
3. If the slopes are equal, compare the y-intercepts.
Open the book to page 192 and read example 3.
Example: Determine whether the lines 3x+ 5y = 2 and 3x + 6 = -5y are parallel, intersecting, perpendicular
or coinciding.
3x + 5y = 2: 5y = -3x + 2
y = -3/5 x + 2/5
3x + 6 = -5y: 5y = -3x – 6
y = -3/5 x – 6/5
Slope are the same but y-intercepts are different so the lines are parallel.
White Board Activity:
Practice 3: Determine whether the lines are parallel, intersecting, perpendicular or coinciding.
a. y = 3x + 7, y = -3x – 4
intersecting: they have different slopes
b. y = -1/3 x + 5, 6y = -2x + 12
y = -2x/6 + 12/6 or y = -1/3 x + 2
parallel: they have the same slope and different y-intercepts
c. 2y – 4x = 16, y – 10 = 2(x – 1)
2y = 4x + 16 or y = 4/2 x + 16/2 or y = 2x + 8
y – 10 = 2x – 2 or y = 2x – 2 + 10 or y = 2x + 8
coinciding: they have the same slope and the same y-intercept
Write the equation of a line parallel to a given equation through a given point.
Steps: 1. Solve the given equation for y.
2. Determine its slope.
3. Since parallel lines have the same slope, use this slope for the new equation.
4. Use the point-slope formula. Use the determined slope and the given point.
5. Solve the equation for y to get slope-intercept form if required.
Example: Write the equation of the line parallel to 2x – 3y = 6 through (4, 5).
2x – 3y = 6: -3y = -2x + 6
y = -2/-3 x + 6/-3
y = 2/3 x – 2
Slope = 2/3
y – 5 = 2/3 (x – 4)
y – 5 = 2/3 x – 8/3
y = 2/3 x – 8/3 + 5
y = 2/3 x – 8/3 + 15/3
y = 2/3 x + 7/3
White Board Activity
Practice: Write the equation of a line parallel to a given equation through a given point.
a. y = 3x – 7, (1, 8)
Since the equation is already solved for y. The slope is the coefficient of the x term.
Slope = 3
y – 8 = 3(x – 1)
y – 8 = 3x – 3
y = 3x – 3 + 8
y = 3x + 5
b. 3x + 4y = 12, (4, -3)
First solve the equation for y.
4y = 12 – 3x
y = 12/4 – 3x/4
y=3–¾x
The slope is the coefficient of the x term.
Slope = -3/4
y + 3 = -3/4 (x – 4)
y + 3 = -3/4 x + (-3/4)(-4/1)
y + 3 = -3/4 x + 12/4
y + 3 = -3/4 x + 3
y = -3/4 x + 3 – 3
y = -3/4 x
Write the equation of the line perpendicular to the given line through the given point.
Steps: 1. Solve the given equation for y.
2. Determine its slope.
3. Perpendicular lines have the opposite-reciprocal slopes.
Determine the slope for the new equation.
4. Use the point-slope formula. Use the determined slope and the given point.
5. Solve the equation for y to get slope-intercept form if required.
Example: Write the equation of the line parallel to 2x – 3y = 6 through (4, 5).
From above the slope is 2/3. Use -3/2 in the equation.
y – 5 = -3/2 (x – 4)
y – 5 = -3/2 x + 12/2
y – 5 = -3/2 x + 6
y = -3/2 x + 6 + 5
y = -3/2 x + 11
White Board Activity:
Practice: Write the equation of the line perpendicular to the given line through the given point.
a. y = -3/2 x + 7, (6, 0)
Since the equation is already solved for y. The slope is the coefficient of the x term.
Slope = -3/2 so use its opposite-reciprocal. Slope = 2/3.
y – 0 = 2/3(x – 6)
y = 2/3 x + (2/3)(-6/1)
y = 2/3 x + (-12/3)
y = 2/3 x – 4
b. 4x – 2y = 8, (-5, 4)
Solve the equation for y. -2y = 8 – 4x
y = 8/-2 – 4x/-2
y = -4 + 2x
Slope = 2 so use its opposite-reciprocal. Slope = -1/2.
y – 4 = -½(x + 5)
y – 4 = -1/2x + (-1/2)(5/1)
y – 4 = -1/2 x – 5/2
y = -1/2 x – 5/2 + 4/1
y = -1/2 x – 5/2 + 8/2
y = -1/2 x + 3/2
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 194 – 195 prob. 8 – 11, 19 – 22, 33 – 40.
For a Grade:
Text: pgs. 194 – 195 prob. 22, 36, 38.