3-5 Slopes of Lines Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

Warm Up Find the value of m. 1. 2. 3. 4. undefined

Objectives Find the slope of a line. Use slopes to identify parallel and perpendicular lines.

Vocabulary rise run slope

The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

Example 1A: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AB Substitute (–2, 7) for (x1, y1) and (3, 7) for (x2, y2) in the slope formula and then simplify.

Example 1B: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AC Substitute (–2, 7) for (x1, y1) and (4, 2) for (x2, y2) in the slope formula and then simplify.

Example 1C: Finding the Slope of a Line Use the slope formula to determine the slope of each line. AD Substitute (–2, 7) for (x1, y1) and (–2, 1) for (x2, y2) in the slope formula and then simplify. The slope is undefined.

A fraction with zero in the denominator is undefined because it is impossible to divide by zero. Remember!

Example 1D: Finding the Slope of a Line Use the slope formula to determine the slope of each line. CD Substitute (4, 2) for (x1, y1) and (–2, 1) for (x2, y2) in the slope formula and then simplify.

Check It Out! Example 1 Use the slope formula to determine the slope of JK through J(3, 1) and K(2, –1). Substitute (3, 1) for (x1, y1) and (2, –1) for (x2, y2) in the slope formula and then simplify.

One interpretation of slope is a rate of change One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

Example 2: Transportation Application Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line. Use the points (4, 260) and (7, 455) to graph the line and find the slope.

Example 2 Continued The slope is 65, which means Justin is traveling at an average of 65 miles per hour.

Check It Out! Example 2 What if…? Use the graph below to estimate how far Tony will have traveled by 6:30 P.M. if his average speed stays the same. Since Tony is traveling at an average speed of 60 miles per hour, by 6:30 P.M. Tony would have traveled 390 miles.

If a line has a slope of , then the slope of a perpendicular line is . The ratios and are called opposite reciprocals.

Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear. Caution!

Example 3A: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3) The products of the slopes is –1, so the lines are perpendicular.

Example 3B: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

Example 3C: Determining Whether Lines Are Parallel, Perpendicular, or Neither Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3) The lines have the same slope, so they are parallel.

Check It Out! Example 3a Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3) Vertical and horizontal lines are perpendicular.

Check It Out! Example 3b Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. KL and MN for K(–4, 4), L(–2, –3), M(3, 1), and N(–5, –1) The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

Check It Out! Example 3c Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. BC and DE for B(1, 1), C(3, 5), D(–2, –6), and E(3, 4) The lines have the same slope, so they are parallel.

Lesson Quiz 1. Use the slope formula to determine the slope of the line that passes through M(3, 7) and N(–3, 1). m = 1 Graph each pair of lines. Use slopes to determine whether they are parallel, perpendicular, or neither. 2. AB and XY for A(–2, 5), B(–3, 1), X(0, –2), and Y(1, 2) 4, 4; parallel 3. MN and ST for M(0, –2), N(4, –4), S(4, 1), and T(1, –5)