1. 3.6 Parallel Lines in a Coordinate Plane Geometry Mrs. Spitz Fall 2005

2. Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Find slopes of lines and use slope to identify parallel lines in a coordinate plane.  Write equations of parallel lines in a coordinate plane.

3. Assignment:  Pgs. 168-170 #4-22; 24-44

4. Slope of parallel lines  In algebra, you learned that the slope of a nonvertical line is the ratio of the vertical change (rise) to the horizontal change (run).  If the line passes through the points (x 1, y 1 ) and (x 2, y 2 ), then the slope is given by slope = rise run m = y 2 – y 1 x 2 – x 1 Slope is usually represented by the variable m.

5. Ex. 1: Finding the slope of train tracks  COG RAILWAY. A cog railway goes up the side of Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section? slope = rise = 4 feet =.4 run 10 feet

6. Ex. 2: Finding Slope of a line  Find the slope of the line that passes throug the points (0,6) and (5, 2).  m = y 2 – y 1 x 2 – x 1 = 2 – 6 5 – 0 = - 4 5

7. Postulate 17 Slopes of Parallel Lines  In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. k1k1 k2k2 Lines k 1 and k 2 have the same slope.

8. Ex. 3 Deciding whether lines are parallel  Find the slope of each line. Is j 1 ║j 2 ? M 1 = 4 = 2 2 M 2 = 2 = 2 1 Because the lines have the same slope, j 1 ║j 2.

9. Ex. 4 Identifying Parallel Lines M 1 = 0-6 = -6 = -3 2-0 2 M 2 = 1-6 = -5 = -5 0-(-2) 0+2 2 M 3 = 0-5 = -5 = -5 -4-(-6) -4+6 2 k3k3 k2k2 k1k1

10. Solution: Compare the slopes. Because k 2 and k 3 have the same slope, they are parallel. Line k 1 has a different slope, so it is not parallel to either of the other lines.

11. Writing Equations of parallel lines  In algebra, you learned that you can use the slope m of a non-vertical line to write an equation of the line in slope-intercept form. slopey-intercept y = mx + b The y-intercept is the y-coordinate of the point where the line crosses the y-axis.

12. Ex. 5: Writing an Equation of a Line  Write an equation of the line through the point (2, 3) that has a slope of 5. y = mx + b 3 = 5(2) + b 3 = 10 + b -7= b  Steps/Reasons why Slope-Intercept form Substitute 2 for x, 3 for y and 5 for m Simplify Subtract.

13. Write the equation  Because m = - 1/3 and b = 3, an equation of n 2 is y = -1/3x + 3 This assignment is due next time we meet.