3 4 Find and Use Slopes of Lines

3. 4 Find and Use Slopes of Lines Define the following words: Slope - The rate of change Rise - The change in y Run - The change in x

Key Concept: Slope of lines in the Coordinate Plane • • Negative slope: falls from left to right Positive slope: rises from left to right Zero slope – horizontal Undefined slope: vertical

EXAMPLE 1 Find slopes of lines in a coordinate plane Find the slope of line a and line d. SOLUTION y 2 – y 1 4 – 2 2 Slope of line a: m = x – x = =– 1 = 6– 8 – 2 2 1 y 2 – y 1 4 – 0 = Slope of line d: m = x – x = 2 1 6– 6 which is undefined. 4 0

Postulate 17: Slopes of Parallel Lines • In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. m || n

EXAMPLE 2 Identify parallel lines Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k 1 through (– 2, 4) and (– 3, 0). m 1 = 0– 4 = –– 41 = 4 – 3 – (– 2 ) Find the slope of k 2 through (4, 5) and (1, 3). m 2 = 1 – 5 3– 4 = – 1 = 4

EXAMPLE 2 Identify parallel lines Find the slope of k 3 through (6, 3) and (5, – 2). m 3 = – 2 – 3 5– 6 – 5 = – 1 = 5 Compare the slopes. Because k 1 and k 2 have the same slope, they are parallel. The slope of k 3 is different, so k 3 is not parallel to the other lines.

Postulate 18: Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1.

EXAMPLE 3 Draw a perpendicular line Line h passes through (3, 0) and (7, 6). Graph the line perpendicular to h that passes through the point (2, 5). SOLUTION STEP 1 Find the slope m 1 of line h through (3, 0) and (7, 6). m 1 = 6 – 0 = 6 = 3 7– 3 4 2

EXAMPLE 3 Draw a perpendicular line STEP 2 Find the slope m 2 of a line perpendicular to h. Use the fact that the product of the slopes of two perpendicular lines is – 1. 3 2 m 2 = – 1 m 2 = – 2 3 STEP 3 Slopes of perpendicular lines Multiply each side by 2 3 Use the rise and run to graph the line.

EXAMPLE 4 Standardized Test Practice SOLUTION The rate at which the skydiver descended is represented by the slope of the segments. The segments that have the same slope are a and c. ANSWER The correct answer is D.