2What we Have Learned so far…. m = SlopeWhere lower case m represents the slope for all linear equations
3What we Have Learned so far…. Slope Intercept FormWhere m is the slope and b is the intercept
4What we Have Learned so far…. Point-Slope FormWhere m is the slope and are a point on the line.
5We will use our prior knowledge of Slopes&Slope-Intercept FormTo learn about Parallel and Perpendicular Lines
6Parallel 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.
7Testing if Lines are Parallel Are the lines y = 3x + 2 and 9x – 3y = -6 parallel?First find the slope of y = 3x + 2The slope m = 3Second find the slope of 9x – 3y = - 69x – 9x – 3y = -9x – 6-3y = -9x – 6y = 3x + 2The slope m = 3Since the slopes are equal the lines are parallel.
8Example: Bellow is the graph of two Parallel Lines. The red line is the graph ofy = – 4x – 3and the blue line is the graph ofy = – 4x – 7Because the red line and the blue line have the same slope, they will NEVER intercept. Therefore they are PARALLEL LINES.
9Slope and Parallel Lines If two non-vertical lines are parallel, then they have the same slope. Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form.y = 2x + 1-5-4-3-2-112345(-3, 2)Rise = 2Run = 1y – y1 = m(x – x1)y1 = 2m = 2x1 = -3Because 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 + 6Apply the distributive property.Add 2 to both sides.y = 2x + 8This is the slope-intercept form of the equation.
10Your turn! Practice Testing if Lines are Parallel Are the linesparallel?Since the slopes are differentthe lines are not parallel.parallel?Are the linesSince the slopes are equalthe lines are parallel.
11Your Turn! Practice Constructing Parallel Lines Find the equation of the line going through the point (4,1) andparallel toFind the equation of the line going through the point (-2,7) andparallel to
12Perpendicular 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 =
13Slope and Perpendicular Lines 90°Slope and Perpendicular LinesIf two non-vertical lines are perpendicular, then the product of their slopes is –1.(5/2) • (-2/5) = -1Slopes are negative reciprocals of each otherFind the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0.-2x x + 4To isolate the y-term, subtract 2x and add 4 on both sides.4y = -2x + 4Divide both sides by 4.y = -1/2x + 1Slope is –1/2.The given line has slope –1/2.Any line perpendicular to this line has a slope that is the negative reciprocal, 2.
14Example: Bellow is the graph of two Perpendicular Lines. The red line is the graph ofy = – 2x + 5and the blue line is the graph ofy = – 1/2 x +4Because 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.
15Testing 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 linesare perpendicular.
16A 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:
17Your 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 perpendicularSince the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.
18Your Turn! Constructing Perpendicular Lines Find the equation of a line going through the point (3, -5) andperpendicular toThe slope of the perpendicular line will be m = 3/2 Using the point-slope equation where the slope m = 3/2 andthe point is (3, -5) we get
19Your Turn! Practice Constructing Perpendicular Lines Find the equation of the line going through the point (4,1) andperpendicular toFind the equation of the line going through the point (-2,7) andperpendicular to
20Student ActivityYou will now receive a worksheet. Turn the worksheet in when completed.