Parallel and Perpendicular Lines (4-4) Objective: Write an equation of the line that passes through a given point, parallel to a given line. Write an equation of the line that passes through a given point, perpendicular to a given line.

Parallel Lines Lines in the same plane that do not intersect are called parallel lines (//). Nonvertical parallel lines have the same slope. All vertical lines are parallel.

Parallel Lines You can write an equation of a line parallel to a given line if you know a point on the line and an equation of the given line. First find the slope of the given line. Then, substitute the point provided and the slope from the given line into the point-slope form: y – y1 = m(x – x1). Transform the equation into slope-intercept form: y = mx + b.

Example 1 Write an equation in slope-intercept form for the line that passes through (4, -2) and is parallel to the graph of y = ½ x – 7. m = ½ x1 = 4 y1 = -2 y – y1 = m(x – x1) y + 2 = ½ (x – 4) y + 2 = ½ x – 2 -2 -2 y = ½ x – 4

Check Your Progress Choose the best answer for the following. Write the slope-intercept form of an equation for the line that passes through (2, 3) and is parallel to the graph of y = ½ x – 1. y = -2x + 3 y = ½ x + 3 y = ½ x + 2 y = -2x – 1 y – y1 = m(x – x1) m = ½ x1 = 2 y1 = 3 y – 3 = ½ (x – 2) y – 3 = ½ x – 1 +3 +3

Perpendicular Lines Lines that intersect at right angles are called perpendicular () lines. The slopes of nonvertical perpendicular lines are opposite reciprocals. That is, if the slope of a line is 4, the slope of the line perpendicular to it is –¼. The opposite reciprocal of a/b is – b/a. Their product is -1. Vertical and horizontal lines are perpendicular.

Example 2 The height of a trapezoid is the length of a segment that is perpendicular to both bases. In trapezoid ARTP, AP and RT are bases. Can EZ be used to measure the height of the trapezoid? Explain. mRT = 4/4 = 1 EZ cannot be a height because -7 is not the opposite reciprocal of 1. mAP = 10/10 = 1 mEZ = -7/1 = -7

Example 2 The height of a trapezoid is the length of a segment that is perpendicular to both bases. In trapezoid ARTP, AP and RT are bases. Are the bases parallel? The bases are parallel because they both have the same slope of 1.

Check Your Progress Choose the best answer for the following. The graph shows the diagonals of a rectangle. Determine whether JL is perpendicular to KM. JL is not perpendicular to KM. JL is perpendicular to KM. Cannot be determined. mJL = -5/4 mKM = 5/4

Comparing Slopes You can determine whether the graphs of two linear equations are parallel or perpendicular by comparing the slopes of the lines.

Example 3 Determine whether the graphs of 3x + y = 12, y = 1/3 x + 2, and 2x – 6y = -5 are parallel or perpendicular. Explain. 3x + y = 12 y = 1/3 x + 2 2x – 6y = -5 -3x -3x m = 1/3 -2x -2x y = -3x + 12 -6y = -2x – 5 m = -3 -6 -6 -6 y = 1/3 x + 2 and 2x – 6y = -5 are parallel because they have the same slope. y = 1/3 x + 5/6 m = 1/3 3x + y = 12 is perpendicular to both the others because it is the opposite reciprocal slope.

Check Your Progress Choose the best answer for the following. Determine whether the graphs of y = -2x + 1, x – 2y = -4, and y = 3 are parallel or perpendicular. y = -2x + 1 and x – 2y = -4 are perpendicular. None of the lines are parallel. y = -2x + 1 and y = 3 are perpendicular. None of the lines are parallel. y = -2x + 1 and x – 2y = -4 are parallel. None of the lines are perpendicular. None of the lines are parallel or perpendicular. m = -2 m = 0 x – 2y = -4 -2y = -x – 4 y = ½ x + 2 -x -x -2 -2 -2 m = ½

Perpendicular Lines You can write the equation of a line perpendicular to a given line if you know a point on the line and the equation of the given line.

Example 4 Write an equation in slope-intercept form for the line that passes through (4, -1) and is perpendicular to the graph of 7x – 2y = 3. m = -2/7 x1 = 4 y1 = -1 7x – 2y = 3 y – y1 = m(x – x1) -7x -7x y + 1 = -2/7 (x – 4) -2y = -7x + 3 y + 1 = -2/7 x + 8/7 -2 -2 -2 -1 -1 y = 7/2 x – 3/2 y = -2/7 x + 1/7 m = 7/2

Check Your Progress Choose the best answer for the following. Write an equation in slope-intercept form for the line that passes through (-3, -2) and is perpendicular to the graph of x + 4y = 12. y = ¼ x + 10 y = 4x + 10 y = -4x + 10 y = -¼ x + 10 x + 4y = 12 m = 4 x1 = -3 y1 = -2 -x -x 4y = -x + 12 4 4 4 y = -¼ x + 3 m = - ¼

Summary Parallel Lines Two nonvertical lines are parallel if they have the same slope. AB // CD

Summary Perpendicular Lines Two nonvertical lines are perpendicular if the product of their slopes is -1. EF  GH