Parallel & Perpendicular Lines in the Coordinate Plane Geometry Parallel & Perpendicular Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Goals Find the slope of lines on the coordinate plane. Determine if two lines are parallel. December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Review: Slope Slope = Rise Run Run = 6 (3, 3) Rise =4 (-3, -1) December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel 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. December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Another Example Slope = Rise Run Run = -3 (-1, 3) Rise =3 (2, 0) December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

We can also use the formula. Given two points and The slope is December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Example Find the slope of the line that passes through (9, 12) and (6, -3). December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Postulate 17 Parallel lines have the same slope. We write: m1 = m2 December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Geometry 3.6 Parallel Lines in the Coordinate Plane Summary Slope measures the steepness of a line. Slope is the Rise/Run. Parallel lines have the same slope. December 10, 2018 Geometry 3.6 Parallel Lines in the Coordinate Plane

Perpendicular Lines in the Coordinate Plane Geometry Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Goals Use slope to identify perpendicular lines in a coordinate plane. Write equations of perpendicular lines. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Review Lines are parallel if they have the same slope. Slope is rise/run. Lines with a positive slope rise to the right. Lines with a negative slope rise to the left. Horizontal Lines: slope = 0. Vertical Lines: slope is undefined (or none). December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Problem Slopes Find the slope of the line containing (4, 6) and (2, 6). Do it graphically: (2, 6) (4, 6) Horizontal Lines have the form y = c. y = 6 December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Problem Slope Find the slope of the line containing (4, 6) and (4, 3). Do it graphically: (4, 6) (4, 3) undefined (no SlopE) x = 4 vertical Lines have the form x = c. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Postulate 18 Two lines are perpendicular iff the product of their slopes is –1. Algebraically: m1 • m2 = –1 A vertical and a horizontal line are perpendicular. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Example m1 2 1 -1 2 m1  m2 m2 December 10, 2018 Geometry 3.7 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! December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

lines are perpendicular. If the product of the slopes is -1, then the lines are perpendicular. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Exception Slope of m1 is ?  Undefined Slope of m2 is ?  Zero m1  m2  –1. But m1  m2! m1 (2, 2) m2 (-2, 1) (3, 1) (2, -1) A vertical line and a horizontal line are perpendicular. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Another way to think of it: Two lines are perpendicular if one slope is the negative reciprocal of the other. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Slope Intercept form review y = mx + b m is the slope b is the y-intercept The y-intercept is at (0, b) Lines are parallel if they have the same slope. They are perpendicular if the product of their slopes is –1. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

More challenging problem These equations are in General Form Ax + By = C Slope is always: December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane Why is this so? Consider the equation: 8x – 4y = 12 Move the 8x: – 4y = – 8x + 12 Divide by –4: y = 2x – 3 Slope is? 2 Now use –A/B: -8/(-4) = 2 December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

The slopes are negative reciprocals, so the lines are perpendicular. For –3x + 2y = 2, slope is For 2x + 3y = –2, slope is The slopes are negative reciprocals, so the lines are perpendicular. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane

Geometry 3.7 Perpendicular Lines in the Coordinate Plane In summary Two lines are parallel if they have the same slope. Two lines are perpendicular if the product of their slopes is –1. General form is Ax + By = C and the slope in this form is –A/B. December 10, 2018 Geometry 3.7 Perpendicular Lines in the Coordinate Plane