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U6 – W4 Parallel and Perpendicular Lines Worksheet 1. Graph each pair of lines on the same coordinate grid. Find their slopes and conclude whether the lines are parallel, perpendicular, or neither. a) 𝑦 = 2𝑥 + 5 𝑎𝑛𝑑 4𝑥 − 2𝑦 + 6 = 0 b) 𝑦 = 1 𝑥 2 − 4 𝑎𝑛𝑑 𝑥 + 2𝑦 + 2 = 0 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 c) 𝑦 = 3 𝑎𝑛𝑑 𝑥 = −2 6 8 10 d) 𝑦 = 3𝑥 𝑎𝑛𝑑 𝑥 + 3𝑦 − 6 = 0 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 4 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 4 6 8 10 2. The slopes of two lines are given. Conclude whether the lines are parallel, perpendicular, or neither. Justify your answers. 2 4 3 −4 a) 𝑚 = 𝑎𝑛𝑑 𝑚 = b) 𝑚 = 𝑎𝑛𝑑 𝑚 = c) 𝑚 = 2 𝑎𝑛𝑑 𝑚 = −2 3 6 d) 𝑚 = 1 𝑎𝑛𝑑 𝑚 = −1 4 e) 𝑚 = 1 5 3 𝑎𝑛𝑑 𝑚 = 0.2 1 4 f) 𝑚 = 2 4 𝑎𝑛𝑑 𝑚 = − 9 3. What is the slope of a line that is parallel to each line? 3 a) 𝑦 = 5 𝑥 − 2 b) 𝑦 = −𝑥 + 7 c) 2𝑥 − 𝑦 + 3 = 0 mparallel = _______ mparalllel = _________ mparallel = ________ mperpendicular = ________ mperpendicular = _________ mperpendicular = _________ d) 4𝑥 + 3𝑦 = 12 mparallel = _______ mperpendicular = ________ e) y = 2 mparalllel = _________ mperpendicular = _________ f) x = -5 mparallel = ________ mperpendicular = _________ 4. Write the equations of two lines that are parallel to the line 3x - 6y - 5 = 0. 5. Write the equations of two lines that are perpendicular to the line 4x + y - 2 = 0. 6. In each of the following, line k and l pass through the given points. Determine if k and l are parallel, perpendicular, or neither. (HINT: Compare the slopes of k and l a) k: (2, 3) & (4, 4); l (3, 6) & (-7, 1) b) k: (2, 5) & (4, 11); l (0, 4) & (-9, 7) c) k: (4, 3) & (6, 7); l (-2, 1) & (0, 0) d) k: (-3, -7) & (9, -7); l (3, 5) & (-1, 4) e) k: (5, 8) & (5, -2); l (-2, 5) & (-2, -1) f) k: (-5, -1) & (-1, 5); l (3, 4) & (7, -2) 10 7. A triangle has vertices A(1, 2), B(3, 8), and C(6, 7). a) Plot these points and draw the triangle. 8 b) Does this appear to be a right triangle? Explain. 4 6 2 c) Find the slopes of the line segments that form this triangle. -10 -8 -6 -4 -2 2 4 6 -2 -4 -6 -8 -10 d) Explain how the slopes can be used to conclude whether or not this is a right triangle. Is it? 8. Determine whether or not the following sets of points form right triangles. Justify your answers with mathematical reasoning. a) A(1, 1), B(-2, 5), C(3, -2) b) P(2, 4), Q(-2, 2), R(5, -2) Answer Key: 1a) parallel b) neither c) perpendicular d) perpendicular 2a) parallel, same slope b) perpendicular, negative reciprocals c) neither d) perpendicular, negative reciprocals e) parallel, same slope f) perpendicular, negative reciprocals 3a) 3/5 and -5/3 b) -1 and 1 c) 2 and -1/2 d) -4/3 and ¾ e) 0 and undefined f) undefined and 0 4. Must have slope = ½ 5. Must have slope of ¼ 6a) parallel b) perpendicular c) perpendicular d) neither e) parallel f) neither 7b) yes, two lines look like they form a 900 angle c) mAB = 3, mBC = -1/3, mAC = 1 d) AB is perpendicular to BC 8a) mAB = -4/3, mBC = -7/5, mAC = -3/2 NO b) mPQ = 1/2, mQR = -4/7, mRP = -2 8 10