6.4 Parallel and Perpendicular Lines Principles of Mathematics 9, pages 326-329 A 1. Graph each pair of lines on the same coordinate grid using pencil and paper or 3. Graph each pair of lines on the same technology. Find their slopes and coordinate grid using pencil and paper or conclude whether the lines are parallel, technology. Find their slopes and perpendicular, or neither. conclude whether the lines are parallel, perpendicular, or neither. a) y = 2r + 3 y = 2x - 1 a) xty=3 xty =2 b) y = 4x + 2 y = - -x+1 b) 3x+ 2y-6=0 2x - 3y + 6=0 4 c) 2x+y-1=0 c) y= 3x + 1 "x+y-2=0 V = . x+1 3 d) xty -2=0 x-y-2=0 d) y= =x+1 - X 4. The slopes of two lines are given. e) y=x+1 y= -x+1 Conclude whether the lies are parallel, f) y = 3x - 2 y = 2x - 3 perpendicular, or neither, Justify your answers. 2. Graph each pair of lines on the same a) m= -, m coordinate grid using pencil and paper or technology. Find their slopes and conclude whether the lines are parallel, b) m=3, m= perpendicular, or neither. c) m = 5, m=-5 a) y = 3 y = -2 b) y=1 x = -1 d) m = 0.4, m= c) y =2 y = X e) m=2-, m= UIN d) x =3 x=0 e) y= x+2 V= -X () m= -- -, m= f) x= 4 2 ' J=-x+4 g) m = 0, m = undefined Chapter 6 . NEL 1055. What is the slope of a line that is parallel to each line? 9. a) Write the line 5x - 2y + 4 = 0 in the form y = mx + b. a) y = 3x + 5 b) State the slope of the line b) y= -2x + 3 5x - 2y + 4 =0. c) State the slope of a line that is c) y= =x+4 perpendicular to the line 5x - 2y + 4 = 0. d) y= - d) Write the equations of two lines 5 that are perpendicular to the line e) 2r + 3y = 12 5x - 2y + 4 =0. () 5x - 3y - 15 =0 10. The lines in the following table are g) x = 3 parallel to the line 2x + 3y = 18. h) y = -4 a) Determine the x- and y-intercepts of the line 2x + 3y = 18. b) Complete the following table. 6. For each line in question 5, give the Line slope of a perpendicular line. Equation x-intercept |y-intercept 2x + 3y = 12 7. Copy and complete the following table. 2x + 3y = 6 Slope of 2x + 3y = -6 Slope of a Slope of a Parallel Perpendicular 2x + 3y = -12 a Line Line Line 2x + 3y = -18 4 c) Describe how you can use intercepts -3 to find a line that is parallel to a given line. OWIN 11. Determine whether or not the following undefined sets of points form right triangles. Justify your answers with mathematical B. a) Write the line 3x + 2y - 7 =0 in reasoning. the form y = mx + b. a) A(1, 3), B(5, 1), C(6, 3) b) State the slope of the line b) D(-2, 5), E(2, 3), F(3, -2) 3x + 2y - 7 = 0. c) M(-4, 2), N(-1, 4), O(1, 1) c) State the slope of a line parallel to the line 3x + 2y - 7 = 0. 12. ALMN has vertices L(-1, 2) and d) Write the equations of two lines M(-4, -1). that are parallel to the line a) Find the coordinates of N such that 3x + 2y - 7 =0. ALMN is a right triangle. b) Is there more than one solution? Explain. 106 NEL . Exercise and Homework Book6.5 Find an Equation for a Line Given the Slope and a Point Principles of Mathematics 9, pages 330-337 A 1. Find the equation of a line with the given 4. Find the equation of a line. slope and passing through the given point. a) parallel to y = - x +3, passing 2 a) m = 2, P(4, 5) through the origin b) m=-4, P(-3, -2) b) perpendicular to y = --x+3, c) m = 2, P(5, -1) passing through (-2, -3) c) parallel to y = -2x + 3, passing d) m= --, P(2, 6) through (0, 0) d) perpendicular to y = 3x + 4, passing through (0, 0) 2. Find the equation of a line with the given slope and passing through the given point. B a) m = 0, P(5, -4) 5. Find an equation for the line parallel to 3x + 5y - 4 =0, with the same b) m = 3, P 2.1 x-intercept as 2x - 3y - 6 = 0. c) m= 2, P(0, 0) 6. Find an equation for the line perpendicular to 2x + 5y - 3 =0, with d) m = _, P(-3, -4) the same y-intercept as 2x + 3y + 6 =0. 7. In Ottawa, you can ride on a tour bus for 3. Find the equation of a line a fixed price plus a variable amount that a) with a slope of 5, passing through depends on the length of the trip. The (2, 3) variable cost is $2/km and a 20-km trip b) with a slope of -4, passing through costs $55. (-3, 5) a) Determine the equation relating cost, c) parallel to y = 2x + 5, passing C, in dollars, and distance, d, in through (3, 2) kilometres. d) perpendicular to y = 3x - 4, passing b) Use your equation to find the cost of through (5, -3) a 15-km tour. " e) parallel to y = 4, passing through c) Graph the relation. (2, 3) d) Use the graph to find the cost of a f) perpendicular to y = -2, passing 15-km tour. through (-3, 1) Chapter 6 . NEL 1076.6 Find an Equation for a Line Given Two Points Principles of Mathematics 9. pages 338-343 1. Find an equation for each line. 2. Find an equation for the line passing through each pair of points. (3, 4) a) A(3, 4) and B(6, 10) b) D(1, 5) and E(3, -3) (1. 2) c) M(-3, 6) and N(1, -4) d) P(-4, 7) and Q(2, -3) b) B 3. a) Find an equation for a line with an 2 x-intercept of -3 and a y-intercept (-2, 2). of 5. b) Find an equation for a line with an x-intercept of 4 and a y-intercept of -2. c) 4. a) Find an equation for a line with the same x-intercept as the line 2x + 5y - 4 =0 and the same (-2,3) 2 4. (3. 3 ) y-intercept as the line 3x - 2y + 8 = 0. -2- b) Find an equation for a line with the same x-intercept as the line 3x - 4y + 6 =0 and the same y-intercept as the line 4x - 5y - 10 = 0. d) (1, 2) -4 -2 0 .2 -24 1(1. -2) -4 Chapter 6 . NEL 109