Pearson Unit 1 Topic 3: Parallel and Perpendicular Lines 3-8: Slopes of Parallel and Perpendicular Lines Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (2)(C) Determine an equation of a line parallel or perpendicular to a given line that passes through a given point. (1)(B) Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.

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

Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear. Coincident lines occupy the same path. Caution! Lines that intersect but are not perpendicular are called oblique lines.

Example: 1 Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, oblique, or coincident. UV given U(0, 2), V(–1, –1) XY given X(3, 1), Y(–3, 3) The products of the slopes is –1, so the lines are perpendicular.

Example: 2 Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, oblique, or coincident. GH given G(–3, –2), H(1, 2) IJ given 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. The lines are oblique.

Example: 3 Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, oblique, or coincident. CD given C(–1, –3), D(1, 1) EF given E(–1, 1), and F(0, 3) The lines have the same slope, so they are parallel.

Example: 4 WX given W(3, 1), X(3, –2) YZ given Y(–2, 3), Z(4, 3) Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, oblique, or coincident. WX given W(3, 1), X(3, –2) YZ given Y(–2, 3), Z(4, 3) Vertical and horizontal lines are perpendicular.

Example: 5 Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, oblique, or coincident. KL given K(–4, 4), L(0, –2) MN given M(-2, 1), and N(2, –5) The slopes are the same, but all four points are on the same line. The lines are coincidental lines.

Given: y = ½ x + 2 and point (-1, 6) Example #6: Write an equation of the line parallel to the given line that contains the given point. Given: y = ½ x + 2 and point (-1, 6) Solution: The lines must have the same ½ slope. Substitute into point-slope form: y – 6 = ½ (x – (-1)) y - 6 = ½ (x + 1) is the point-slope form. If slope-intercept form is requested: Distribute the ½: y – 6 = ½ x + ½ Add 6 to both sides: y = ½ x + 6.5

Given: y = ½ x + 2 and point (-1, 6) Example #7: Write an equation of the line perpendicular to the given line that contains the given point. Given: y = ½ x + 2 and point (-1, 6) Solution: The lines must have -2 slope. Substitute into point-slope form: y – 6 = -2(x – (-1)) y – 6 = -2(x + 1) is the point-slope form. If slope-intercept form is requested: Distribute the -2 : y – 6 = -2x - 2 Add 6 to both sides: y = -2x + 4

EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW. ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

Example: 8 Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither. BC given B(1, 1), C(3, 5) DE given D(–1, –4), E(3, 4) The lines have the same slope, so they are parallel.