Lesson 4.6 Objectives Determine if two lines are parallel or perpendicular. Write equations of parallel and perpendicular lines. Slopes of Parallel and Perpendicular Lines Parallel and perpendicular lines are all around us. Both parallel and perpendicular lines are often found in transportation. For example, railroad tracks are an example of parallel lines. The intersections of roadways are often examples of perpendicular lines. Parallel Lines Parallel lines are lines that never intersect and are always the same distance apart. Activity 1 Parallel Lines 1 Graph each of the following equations on the same Cartesian coordinate system. see margin oordinate syste a. y 1 x 1 b. y 1 x 5 c. y 1 x 6 e margin. What relationship appears to exist among the graphs of the three equations? They are parallel. What relationship exists among m the equations? The slopes are equal; they are all 4 Complete the following conjecture. Lines that are parallel have slopes. equal Slopes of Parallel Lines Two nonvertical lines are parallel if they have the same slope and different y-intercepts. All vertical lines are parallel. 1. 8 Chapter 4 Linear Equations
Example 1 Finding the Equation of a Parallel Line te th Write the equation of a line that is parallel to the graph of y contains the point ( 6, 1). Solution Because the equation is parallel to y x 4, the slope will be. x 4 and Use the slope and the point ( 6, 1) to find the y-intercept of the equation. Substitute these values into the slope-intercept formula. y mx b 1 ( 6) b 1 4 b 1 4 4 4 b 5 b Use the slope and y-intercept (0, 5) to write the equation of the line in slope intercept form. y x 5 To check your equation, you can graph both lines on the same Cartesian coordinate system to verify that they are parallel. y x 5 y x 4 Ongoing Assessment Write the equation of a line that is parallel to the graph of y x 7 and contains the point (, ). Perpendicular Lines 4.6 Slopes of Parallel and Perpendicular Lines 9
Perpendicular lines are lines that intersect to form right angles. Recall that right angles are angles measuring 90. Activity Perpendicular Lines 1 Graph both of the following equations on the same Cartesian coordinate system. see margin a. y x b. y 1 x 4 What relationship appears to exist among the graphs of the equations? They are perpendicular. Graph both of the following equations on the same Cartesian coordinate system. see margin a. y 4 x e margin. b. y 4 x 5 4 What relationship appears to exist among the graphs of the equations? They are perpendicular. 5 Examine the pairs of equations in Steps 1 and. What relationship exists among each pair of equations? The slopes have a product of 1, or are negative reciprocals. 6 Complete the following conjecture for non-vertical and non-horizontal lines. The slopes of lines that are perpendicular have a product of. 1 Slopes of Perpendicular Lines Two lines are perpendicular if their slopes are negative reciprocals. A vertical line and a horizontal line are perpendicular. Negative reciprocals are two numbers that have a product of 1. For example, and 1 are negative reciprocals. 1 1 Ongoing Assessment Find the negative reciprocal of each number. a. 1 1 b. c. 5 5 40 Chapter 4 Linear Equations
Example Equation of a Perpendicular Line An architect is drawing a blueprint of a house on grid paper. She draws each view of the house separately. On the side-view of the house, the roof will meet at a 90 angle. One side of the roof lies on the graph of the equation y x 7. The other side of the roof lies on the graph of an equation that contains the point (4, 9). What is the equation of this line? Solution First find the slope of the other side of the roof. The side of the roof that lies on the line y x 7 has a slope of 1. The negative reciprocal of 1 is 1. Then use the point (4, 9) to find the y-intercept of the line. y 1x b 9 1(4) b 9 4 b 9 4 4 4 b 7 b Therefore, the equation of the line is y x 7. Ongoing Assessment a. Write the equation of a line that is perpendicular to the graph of x y 9 and contains the point ( 8, ). y x 15 b. Write the equation of a line parallel to the graph of 4x 1 5y 9 and contains the point (5, ). y 4 5 x 4.6 Slopes of Parallel and Perpendicular Lines 41
Lesson Assessment Think and Discuss 1. Given parallel lines, explain the relationship of their slopes.. Given perpendicular lines, explain the relationship of their slopes.. Explain the meaning of negative reciprocals and give an example. 4. Explain how to write the equation of a line that is parallel to the graph of the equation y 1 x 5. 5. Describe how to find the equation of a line that is perpendicular to a given line and contains a given point. Practice and Problem Solving Determine if the graphs of each pair of equations are parallel, perpendicular, or neither. 6. y 4x + 7. y x 5 parallel y 1 4 x 10 y 4 x 4 8. y x 6 parallel 9. y x + 11 neither 5 x 5y 5 x y 16 10. y 6 x 14 perpendicular 11. x 8y 4 parallel 5 10x 1y 7 y 1 4 x Write the equation of each line that is parallel to the given line and contains the given point. see margin 1. y x 4; ( 1, 1) 1. y 1 x 1; (6, 5) 14. y 5 x 10; ( 5, 4) 15. y x ; (14, 8) 7 16. x y 1; ( 11, 17) 17. x 4y 8; (1, 9) Write the equation of each line that is perpendicular to the given line and contains the given point. see margin 18. y 1 x 1; (0, 4) 19. y x 5; (, ) 0. y 4x ; ( 1, 6) 1. y x ; (5, 8). x 5y 0; (, 0). 8x 14y 14; (, 8) 4 Chapter 4 Linear Equations
Graph each set of points to draw the quadrilateral. Then determine if the quadrilateral is a parallelogram. 4. A(, 4), B(4, 1), C(, 1), D( 5, 4) 5. E(0, 5), F(5, ), G( 1, 4), H( 5, 4) 6. J(, 8), K(1, 0), L( 4, ), M(, 6) Graph each set of points to draw the quadrilateral. Then determine if the quadrilateral is a rectangle. 7. N(, 4), O(, ), P( 6, ), Q( 6, 4) 8. R( 5, 7), S(, 10), T(5, ), U(, 0) 9. W( 6, 9), X( 8, 1), Y(, 1), Z(0, 6) Mixed Review 0. One light year is the distance light travels in one year. Light travels about 5.88 10 1 miles in one year. Our galaxy is approximately 100,000 light years in diameter. What is the diameter, in miles, of our galaxy? Write your answer in scientific notation. Write and simplify an algebraic expression that fits the given information for each problem. 1. two more than three times a number. half of the difference four times between ten times a number and four times the number 1 (10x 4x) or x. In a certain card game, 9 rounds are played using several combined decks of cards. In a round, each player is dealt two cards plus x cards for the round of play. How many cards are dealt in round x to p players? Solve each equation for c. 4. 7c 8 48 5. c 5 17 6. 11 8 c 7. 4 c 8. 0% of c 10 9. c% of 90 150 1 4.6 Slopes of Parallel and Perpendicular Lines 4