HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 3.4: Parallel and Perpendicular Lines

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Slopes of parallel lines. o Slopes of perpendicular lines.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Slopes of Parallel Lines o The slope of a line is a precise indication of its “steepness”, and two lines are parallel if and only if they have the same slope. In the case of vertical lines, this means they both have undefined slope. o This fact is clear from the formula for slope: two lines are parallel if and only if they both “rise” vertically the same amount relative to the same horizontal “run”. o We can use this observation to derive equations for lines that are described in terms of other lines.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Slopes of Parallel Lines Find the equation, in slope-intercept form, for the line which is parallel to the line and which passes through the point. Step 1: Write equation in slope-intercept form. Step 2: Use point-slope form. Step 3: Solve for to obtain slope-intercept form. Use slope to write a new equation that passes through the point.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Slopes of Parallel Lines Determine if the quadrilateral (four sided figure) whose vertices are (-2,0),(3,-1),(4,1), and (-1,2) is a parallelogram (a quadrilateral in which both pairs of opposite sides are parallel). The slopes of the left and right sides are, respectively, and The slopes of the top and bottom sides are, respectively, and The figure is a parallelogram.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Parallel Lines Two non-vertical lines with slopes and are parallel if and only if Also, two vertical lines (with undefined slopes) are always parallel to each other.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular Lines o The relationship between the slopes of perpendicular lines is a bit less obvious. o Consider a non-vertical line, and consider two points and on the line, as shown on the next slide. o These two points can be used, of course, to calculate the slope of, with the result that

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular Lines If we now draw a line perpendicular to, we can use and to determine the slope of line. There are an infinite number of lines that are perpendicular to ; one of them is drawn on the next slide.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular Lines Note that in rotating the line by to obtain, we have also rotated the right triangle drawn with dashed lines.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular Lines o However, to travel along the line from the point to the second point drawn requires a positive rise and a negative run, whereas the rise and run between and are both positive. o In other words,, the negative reciprocal of the slope. o This relationship always exists between the slopes of two perpendicular lines, assuming neither one is vertical.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular Lines Suppose and represent the slopes of two lines, neither of which is vertical. The two lines are perpendicular if and only if (equivalently ). Vertical lines (undefined slope) and horizontal lines (zero slope) are also perpendicular to each other.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. The Slopes of Perpendicular lines Important Parallel lines have the same slope. For example: Perpendicular lines have slopes that are negative reciprocals of each other. For example:

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Slopes of Perpendicular Lines Find the equation, in standard form, of the line that passes through the point and that is perpendicular to the line. The line is a horizontal line, and hence any line perpendicular to it must be a vertical line.

HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Slopes of Perpendicular Lines Prove that the two lines and are perpendicular to each other. The easiest way to do this is to rewrite each equation in slop-intercept form. Since the slope of the second line is the negative reciprocal of the slope of the first line, these two lines are perpendicular to one another.