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Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines. What we Have Learned so far…. m = Slope. Where lower case m represents the slope for all linear equations. What we Have Learned so far…. Slope Intercept Form. Where m is the slope and b is the intercept.
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What we Have Learned so far…. m = Slope Where lower case m represents the slope for all linear equations
What we Have Learned so far…. Slope Intercept Form Where m is the slope and b is the intercept
What we Have Learned so far…. Point-Slope Form Where m is the slope and are a point on the line.
We will use our prior knowledge of Slopes & Slope-Intercept Form To learn about Parallel and Perpendicular Lines
Parallel Lines What are Parallel Lines? Two lines with the Same Slope are said to be Parallel lines. When two Parallel Lines are graph they will Never intersect. We can decide algebraically if two lines are Parallel by finding the slope of each line and seeing if the Slopes are Equal to each other.
Testing if Lines are Parallel Are the lines y = 3x + 2 and 9x – 3y = -6 parallel? First find the slope of y = 3x + 2 The slope m = 3 Second find the slope of 9x – 3y = - 6 9x – 9x – 3y = -9x – 6 -3y = -9x – 6 y = 3x + 2 The slope m = 3 Since the slopes are equal the lines are parallel.
Example:Bellow is the graph of two Parallel Lines. The red line is the graph of y = – 4x – 3 and the blue line is the graph of y = – 4x – 7 Because the red line and the blue line have the same slope, they will NEVER intercept. Therefore they are PARALLEL LINES.
y –y1 =m(x –x1) 5 y = 2x + 1 4 (-3, 2) 3 Run = 1 2 1 Rise = 2 y1 = 2 m= 2 x1 = -3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -2 -3 -4 -5 Slope and Parallel LinesIf two non-vertical lines are parallel, then they have the same slope. Write an equation of the line passing through(-3, 2)and parallel tothe line whose equation isy =2x + 1. Express the equation in point-slope form and y-intercept form. Because the slope of the given line is 2, m = 2 for the new equation. y –2=2[x –(-3)] y – 2 =2(x + 3) y – 2 = 2x + 6 Apply the distributive property. + 2 + 2 Add 2 to both sides. y = 2x + 8 This is the slope-intercept form of the equation.
Your turn!Practice Testing if Lines are Parallel Are the lines parallel? Since the slopes are different the lines are not parallel. parallel? Are the lines Since the slopes are equal the lines are parallel.
Your Turn!Practice Constructing Parallel Lines Find the equation of the line going through the point (4,1) and parallel to Find the equation of the line going through the point (-2,7) and parallel to
Perpendicular Lines What are Perpendicular Lines? Perpendicular lines are lines that intersect forming a right angle (90˚). Perpendicular lines are lines that have exact opposite slopes. We can decide algebraically if two lines are perpendicular by finding the slope of each line and seeing if the slopes are negative reciprocals of each other. This is equivalent to multiplying the two slopes together and seeing if their product is –1. If one line has the slope ‘m’ then a perpendicular line to that line will have a slope = .
90° 4 4 Slope is –1/2. • Slope and Perpendicular Lines • If two non-vertical lines are perpendicular, then the product of their slopes is –1. • (5/2) • (-2/5) = -1 • Slopes are negative reciprocals of each other Slope and Perpendicular Lines Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0. -2x + 4 -2x + 4 To isolate the y-term, subtract 2x and add 4 on both sides. 4y = -2x + 4 Divide both sides by 4. y =-1/2x+ 1 The given line has slope –1/2. Any line perpendicular to this line has a slope that is the negative reciprocal, 2.
Example:Bellow is the graph of two Perpendicular Lines. The red line is the graph of y = – 2x + 5 and the blue line is the graph of y = – 1/2 x +4 Because the red line and the blue line have the exact opposite slope, they will always intercept and form a 90˚ angle. Therefore the lines are PERPENDICULAR.
Testing if Lines Are Perpendicular Are the lines 2x + y = 5 and y = 1/2x + 4 perpendicular? Since the slopes are negative reciprocals of each other the lines are perpendicular.
A Point? A Line? Write an equation of the line passing through (-3,6) and perpendicular to the line whose equation is y=1/3 x +4 Express in point-slope form and slope-intercept form. perpendicular slope:
Your Turn!Practice Testing if Lines Are Perpendicular Since the slopes are not negative reciprocals of each other (their product is not -1) the lines are not perpendicular Since the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.
Your Turn!Constructing Perpendicular Lines Find the equation of a line going through the point (3, -5) and perpendicular to The slope of the perpendicular line will be m = 3/2 Using the point-slope equation where the slope m = 3/2 and the point is (3, -5) we get
Your Turn!Practice Constructing Perpendicular Lines Find the equation of the line going through the point (4,1) and perpendicular to Find the equation of the line going through the point (-2,7) and perpendicular to
Student Activity You will now receive a worksheet. Turn the worksheet in when completed.
