3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines
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1 3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x y = x 3 4. y = 4 x November 2, Proofs with Perpendicular Lines
2 Geometry 3.5 Equations of Parallel and Perpendicular Lines Day 1
3 Essential Question How can you write the equation of a line that is parallel to a given line and passes through a given point?
4 Review: Slope Slope = Rise Run Example 1 Graph the points (-3,-1) and (3,3). Rise =4 Run = 6 (3, 3) Now count out the slope. (-3, -1) Slope
5 Reminder Lines with a positive slope rise to the right. Lines with a negative slope drop to the right. Lines with zero slope are horizontal. Lines with an undefined slope are vertical.
6 Your Turn Slope = Rise Run Graph the points (2,0) and (-1,3). Now count out the slope. (-1, 3) Rise =-3 Run = 3 Slope: m = 3 3 = 1 (2, 0)
7 We can also use the formula. Given two points The slope is x and 1, y 1 x 2, y 2 m y 2 y 1 x 2 x 1
8 Example 2 Find the slope of the line that passes through (9, 12) and (6, -3). 12 ( 3) 15 m m
9 Your Turn Find the slope of the line that passes through (-1, 2) and (1, -4). m 2 ( 4) m ( 1) 6 2 3
10 Slope-Intercept Form To be able to write an equation in slopeintercept form, you must know two things: 1.The slope of the line, m. 2.The y-intercept, b.
11 Equation of a Line Slope-Intercept form: y = mx + b m is the slope b is the y-intercept The y-intercept is the value of y where the line crosses the y-axis. The y-intercept has the coordinate (0, b)
12 Example 3 Write the equation of a line with slope of 3 and y-intercept of 8. y = mx + b y = 3x + 8
13 Example 4
14 Example 5 y = 3 4 x 3
15 Example 6 Write the equation of a line with a slope of 2 that passes through (4, 1). y = mx + b Now solve for b. 1 = -2(4) + b 1 = -8 + b 9 = b y = -2x + 9
16 Your Turn Write the equation of a line that has a slope of 4 and passes through (1, 3). Solution: y = mx + b. m =? y = 4x + b. x =? y =? y = 4x 7 3 = 4(1) + b 7 = b
17 Theorem 3.13 Slopes of Parallel Lines In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
18 Theorem 3.13 Simplified Parallel lines have the same slope. We write: m 1 = m 2
19 Example 7 Write the equation of a line parallel to y = 5x + 10 that has a y-intercept of 6. Think: Parallel Lines = Same Slope y = mx + b m = 5 b = 6 y = 5x 6
20 Example 8 Write the equation of a line parallel to y = x that has a y- intercept of 2. Think: Parallel Lines = Same Slope y = mx + b m = 1 b = 2 y = x+2
21 Summary Slope measures the steepness of a line. Slope is the Rise/Run. Slope intercept form is y = mx + b. The y-intercept is (0, b). Parallel lines have the same slope.
22 Assignment
23 3.5 Day 2 Warm Up Graph each line. (Hint: Solve for y first.) 1. y + 6x = x 2y = x + 3y = x + 3y = 9 November 2, Proofs with Perpendicular Lines
24 Geometry 3.5 Perpendicular Lines in the Coordinate Plane
25 Essential Question How can you write the equation of a line that is perpendicular to a given line and passes through a given point?
26 Review Lines are parallel if they have the same slope. Slope is rise/run. Lines with a positive slope rise to the right. Lines with a negative slope drop to the right.
27 Special Cases Horizontal Line Vertical Line Slope = 0 Equation: y = b Undefined Slope Equation: x = a
28 Write the equation of a line with no slope that passes through (3, 5). x = 3
29 Example 1 Find the slope of the line containing (4, 6) and (2, 6). m Do it graphically: (2, 6) (4, 6) Horizontal Lines have the form y = c. y = 6
30 Example 2 Find the slope of the line containing (4, 6) and (4, 3) m undefined (no Slope) What is the equation of the line? Vertical Lines have the form x = c. Do it graphically: x = 4 (4, 6) (4, 3)
31 Theorem 3.14: Slopes of Perpendicular Lines In the coordinate plane, two nonvertical lines are perpendicular iff the product of their slopes is -1. Algebraically: m 1 m 2 = 1 A vertical and a horizontal line are perpendicular.
32
33 Theorem 3.14 Another Way to Think of It Two lines are perpendicular if one slope is the opposite reciprocal of the other. m = 1 4
34 Example 3 Let s look at perpendicular lines on a graph m 1 1 m 1 2 m m m2 1 m 1 m 2 m 2
35 You don t need a picture. Line A contains (2, 7) and (4, 13). Line B contains (3, 0) and (6, -1). Are the lines perpendicular? m A m B ( 3) 1 3 YES!
36 Your Turn Line A contains (-2, 2) and (0, 2). Line B contains (-2, 3) and (2, 1). Are the lines perpendicular? m A m B YES!
37 Your Turn Again Line A contains (4, -1) and (2, 2). Line B contains (-2, 0) and (4, 3). Are the lines perpendicular? NO! B A m m
38 Exception Slope of m 1 is? (-2, 1) m 1 (2, 2) (3, 1) (2, -1) m 2 Undefined Slope of m 2 is? Zero m 1 m 2 1. But m 1 m 2! A vertical line and a horizontal line are defined as perpendicular.
39 Example 4 Write the equation of a line perpendicular to y = 5x + 10 that has a y-intercept of 10. m = 1 5 b = 10 y = mx + b y = 1 x 10 5
40 Example 5 Write the equation of a line perpendicular to 4x 2y = 8 that has a y- intercept of 4. 2y = 4x + 8 y = 2x 4 m = 1 2 b = 4 y = mx + b y = 1 2 x + 4
41 Example 6 Write the equation of a line perpendicular to 3x 5y = 25 that passes through (6, -4). 3x 5y = 24 5y = 3x + 25 m = 5 3 y = 3 5 x 5 b = 6 y = mx + b 4 = 5 (6) + b 3 4 = 10 + b 6 = b y = 5 3 x + 6
42 Example 7 Determine whether the lines are parallel, perpendicular, or neither. y = 3x + 7 y = 3x 4 m 1 = 3 m 2 = 3 Are they the same? Are they opposite reciprocals? These lines are neither!
43 Example 8 Determine whether the lines are parallel, perpendicular, or neither. 5x + 4y = 12 x 4y = 12 4y = 5x 12 4y = x 12 y= 5 x 3 4 y = 1 4 x + 3 m 1 = 5 4 m 2 = 1 4 These lines are neither!
44 Your Turn Determine whether the lines are parallel, perpendicular, or neither. 2y 4x = 16 y 10 = 2(x 2) 2y = 4x + 16 y = 2x + 8 y 10 = 2x 4 m 1 = 2 m 2 = 2 These lines are parallel! y = 2x + 6
45 Point Slope Form y y 1 = m(x x 1 ) m is the slope. (x 1, y 1 ) is a point on the line.
46 Example 9 For each equation, identify the slope and a point on the line, then graph the line. a. y 2 = 3(x 5) m = 3 (5, 2) b. y + 2 = 2 3 m = 2 3 (x 3) (3, -2)
47 Example 10 Write the equation of a line with slope 6 and passes through (3, -4). y y 1 = m(x x 1 ) y ( 4) = 6(x 3) y + 4 = 6(x 3)
48 Your Turn Write the equation of a line that has a slope of 4 and passes through (1, 3). y y 1 = m(x x 1 ) y ( 3) = 4(x 1) y + 3 = 4(x 1)
49 Example 11 Write the equation of the line through the points ( 3, 2) and (1, 3) y y 1 = m(x x 1 ) y 2 = 1 (x 1) 4 Pick a point: (1, 2) m = 3 2 = 1 1 ( 3) 4
50 In summary Two lines are parallel if they have the same slope. Two lines are perpendicular if the product of their slopes is 1.
51 Assignment
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