Parallel Lines

Section 3.2B – Slope of a Line Chapter 3: Lines Parallel Lines • • • Parallel lines never meet (intersect) and they are always the same distance apart. Parallel lines have the same slope (equal slopes). If 𝑚1 and 𝑚2 are the slopes of two parallel lines, then 𝑚1 = 𝑚2 (This is a symbolic way to say parallel lines have the same slope.) All different vertical lines are parallel to each other. Parallel vertical lines have 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 x − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠. • For Example: • • • • 1 1 The lines: 𝑦 = 2 𝑥 + 5 and 𝑦 = 2 𝑥 + 1 are graphed below. Each line is written in slope-intercept form and the number in front of the x is the slope of each line. 1 The slope of each line is 𝑚 = . 2 The graphs are parallel because each line has the same slope. • • • The lines 𝑥 = −2 𝑎𝑛𝑑 𝑥 = 4 are graphed below Each graph is a vertical line as equations that only have an x have graphs that are vertical lines. The graphs are parallel because both lines are vertical Perpendicular Lines • • • • Perpendicular lines form a right angle (90-degree angle). Slopes of perpendicular lines are negative reciprocals. The product of the slopes of perpendicular lines is -1. Below is a symbolic way to express that lines are perpendicular. If 𝑚1 and 𝑚2 are the slopes of two perpendicular lines, then: 1 o their slopes are negative reciprocals of each other, 𝑚1 = − . 𝑚 2 the product of their slopes is −1: 𝑚1 ∙ 𝑚2 = −1 A vertical line and a horizontal line are always perpendicular to each other. o • For example: • • • • 3 −4 The lines: 𝑦 = 4 𝑥 + 5 and 𝑦 = 3 𝑥 − 6 are graphed below. Each line is written in slope-intercept form and the number in front of the 𝑥 is the slope of each line. 3 4 The slopes are 𝑚 = 4 𝑎𝑛𝑑 𝑚 = − 3 The slopes are negative reciprocals of each other (reciprocals with opposite signs.) The graphs are perpendicular because the slopes are negative reciprocals. • • • The lines 𝑥 = 3.6 𝑎𝑛𝑑 𝑦 = 3 are graphed below The graph of x = 3.6 is a vertical line as the equation only has an x. The graph of y = 3 is a horizontal line as the equation only has a y. The graphs are perpendicular since one line is vertical and the other is horizontal. 2 3 For Example: Given the line: 𝑦 = 𝑥 − 4 a) Find the slope of the given line b) Find the slope of all lines parallel to the given line c) Find the slope of all lines perpendicular to the given line. Answers: 2 a) 𝑚 = 3 (the equation is in slope intercept form, so the number in front of the x is the slope of the line. 2 𝑏) 𝑚 = (the slope must be the same as parallel lines have equal slopes) 3 −3 𝑐) 𝑚 = 2 (the slope needs to be the reciprocal with signs changed as perpendicular lines have slopes that are negative reciprocals.) For Example: Given below are descriptions of two lines. Line 1: Goes through (5, 7) 𝑎𝑛𝑑 (6, 4) Line 2: Goes through (4,12) 𝑎𝑛𝑑 (7,13) Find the slope of Line 1: Find the slope of Line 2: Are the lines parallel, perpendicular or neither? 4−7 Slope of Line 1: 𝑚 = 6−5 = Slope of Line 2: 𝑚 = −3 1 13−12 7−4 1 3 m= 1 3 Lines are perpendicular since the slopes are negative reciprocals. For Example: Given below are descriptions of two lines. Slope of Line l: 𝑚 = 8−(−2) 6−4 Line 1: Goes through (4, −2) 𝑎𝑛𝑑 (6,8) Line 2: Goes through (6,9) 𝑎𝑛𝑑 (9,24) Slope of Line 2: 𝑚 = 24−9 9−6 Find the slope of Line 1: Find the slope of Line 2: Are the lines parallel, perpendicular or neither? = m = -3 = = 10 2 15 3 =5 m=5 =5 m=5 Lines are parallel since they have the same slope. For Example: Given below are descriptions of two lines. Slope of Line 1: 𝑚 = 3−(−4) 2−1 3−10 Line 1: Goes through (1, −4) 𝑎𝑛𝑑 (2,3) Line 2: Goes through (4,10) 𝑎𝑛𝑑 (5,3) Slope of Line 2: 𝑚 = 5−4 = 𝑚 = −7 #35 – 46: a) Find the slope of the given line b) Find the slope of all lines parallel to the given line c) Find the slope of all lines perpendicular to the given line. −3 𝑥 5 +2 37) 𝑦 = 3𝑥 − 2 2 36) 𝑦 = −7 1 = −7 Lines are neither, the slopes are not the same, nor are they reciprocals with different signs. Find the slope of Line 1: Find the slope of Line 2: Are the lines parallel, perpendicular or neither? 35) 𝑦 = 7 =1=7 m=7 −2 𝑥 7 +6 38) 𝑦 = 2𝑥 − 3 7 39) 𝑦 = 9 𝑥 − 4 40) 𝑦 = 5 𝑥 − 1 41) 𝑦 = −6𝑥 + 5 42) 𝑦 = −4𝑥 + 8 43) 𝑥 = 2 44) 𝑥 = −5 45) 𝑦 = 4 46) 𝑦 = 1 # 47 – 52: Given below are descriptions of two lines. Find the slope of Line 1: Find the slope of Line 2: Are the lines parallel, perpendicular or neither? 47) Line 1: Goes through (6, −3) 𝑎𝑛𝑑 (5,9) Line 2: Goes through (17,18) 𝑎𝑛𝑑 (5,17) 48) Line 1: Goes through (1, −2) 𝑎𝑛𝑑 (7,9) Line 2: Goes through (6,8) 𝑎𝑛𝑑 (17,2) 49) Line 1: Goes through (8,2) 𝑎𝑛𝑑 (−6,3) Line 2: Goes through (−5,9) 𝑎𝑛𝑑 (9,8) 50) Line 1: Goes through (4, −2) 𝑎𝑛𝑑 (3, −8) Line 2: Goes through (6,6) 𝑎𝑛𝑑 (9,24) 51) Line 1: Goes through (2, −2) 𝑎𝑛𝑑 (3,1) Line 2: Goes through (10,5) 𝑎𝑛𝑑 (9,8) 52) Line 1: Goes through (7, −14) 𝑎𝑛𝑑 (−2,5) Line 2: Goes through (0,5) 𝑎𝑛𝑑 (9,24)