3.5 Parallel and PerpendicularLines in the Coordinate Plane Geometry 3.5 Parallel and PerpendicularLines in the Coordinate Plane EQ: How do find the equation of lines || and perp to other lines?

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Topic/Objective Find the slope of lines on the coordinate plane. Determine if two lines are parallel or perpendicular. Write the equation of parallel and perpendicular lines. February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Review: Slope Slope = Rise Run Run = 6 (3, 3) Rise =4 (-3, -1) February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Reminder Lines with a positive slope rise to the right. Lines with a negative slope rise to the left. Lines with zero slope are horizontal. Lines with no slope are vertical. February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Another Example Slope = Rise Run Run = -3 (-1, 3) Rise =3 (2, 0) February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

We can also use the formula. Given two points and The slope is February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Example Find the slope of the line that passes through (9, 12) and (6, -3). February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Theorem 3.13 Slopes of parallel lines Parallel lines have the same slope. We write: m1 = m2 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Theorem 3.14 Slopes of Perp lines Two lines are perpendicular iff the product of their slopes is –1. Algebraically: m1 • m2 = –1 A vertical and a horizontal line are perpendicular. February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Example m1 2 1 -1 2 m1  m2 m2 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

You don’t need a picture. Line A contains (2, 7) and (4, 13). Line B contains (3, 0) and (6, -1). Are the lines perpendicular? YES! February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Equation of a Line Slope-Intercept form: y = mx + b m is the slope b is the y-intercept The y-intercept is the value of y where the line crosses the y-axis. The y-intercept has the coordinate (0, b) February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Slope-Intercept Form To be able to write an equation in slope-intercept form, you must know two things: The slope of the line, m. The y-intercept, b. y=mx + b February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Example Write the equation of a line with slope of 3 and y-intercept of 8. y = mx + b y = 3x + 8 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Example Write the equation of a line parallel to y = 5x + 10 that has a y-intercept of –6. Think: Parallel Lines = Same Slope y = mx + b m = 5 b = –6 y = 5x – 6 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Another Example Write the equation of a line parallel to y = x + 1984 that has a y-intercept of 2. Think: Parallel Lines = Same Slope y = mx + b m = 1 b = 2 y = x+2 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Example Write the equation of a line with a slope of –2 that passes through (4, 1). y = mx + b Now solve for b. 1 = -2(4) + b 1 = -8 + b 9 = b y = -2x + 9 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane You try it… Write the equation of a line that has a slope of 4 and passes through (1, –3). Solution: y = mx + b. m = ? y = 4x + b. x = ? y = ? –3 = 4(1) + b –7 = b y = 4x – 7 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Write the equation of a line with no slope that passes through (3, 5). x = 3 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane     Perpendicular slope   6 = 18 + b   -12 = b February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Find the distance from the point (1,0) to the line y = -x + 3 Step I: Find equation line perp to y = -x + 3 passing through (1,0) m = 1 0 = 1(1) + b -1 = b y = 1x - 1 Step II: Now find the point of intersection of the two lines by solving a system of equations. y = -x + 3 y = x – 1 (2,1) 2y = 2 y = 1 1 = x – 1 x = 2 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Find the distance from the point (1,0) to the line y = -x + 3 Step III Now find the distance between (1,0) and (2,1)         February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane Summary Slope measures the steepness of a line. Slope is the Rise/Run. Slope intercept form is y = mx + b. The y-intercept is (0, b). Parallel lines have the same slope. Perpendicular slopes have a product of -1 February 23, 2019 Geometry 3.5 Parallel & Perpendicular Lines in the Coordinate Plane