CHAPTER Slopes of lines

SAT Problem of the day

Solution to the SAT Problem of the day Right Answer: E

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

What is the slope of a line 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.

Slope of a line

Use the slope formula to determine the slope of each line. A) B) C) AB AC AD

Student guided practice Do problems 2-5 in your book page 185

slope 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.

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.

Application 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.

Parallel lines Parallel lines and their slopes are easy. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel.

Perpendicular slopes Perpendicular lines are a bit more complicated. If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will be a decreasing line). So perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down. Put this together with the sign change, and you get that the slope of the perpendicular line is the "negative reciprocal" of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other. In numbers, if the one line's slope is m = 4 / 5, then the perpendicular line's slope will be m = – 5 / 4. If the one line's slope is m = –2, then the perpendicular line's slope will be m = 1 / 2.

Slopes

Example 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)

Example 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)

Example 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)

Student guided practice Do problems 6-9 in your book page 185

Worksheet Lets do some problems in the worksheet

Homework Do problems in your book page 186

Closure Today we learned about finding the slope of a line given two points. Next class we are going to learned about lines in the coordinate plane