Theorem 3.13 Slopes of Parallel Lines

Equations of Parallel and Perpendicular Lines Essential Question: How can you write an equation of a line that is parallel or perpendicular to a given line and passes through a given point?   1. EXPLORATION: Writing Equations of Parallel and Perpendicular Lines Work with a partner. Write an equation of the line that is parallel or perpendicular to the given line and passes through the given point. Use a graphing calculator to verify your answer. What is the relationship between the slopes?   a. coordinate plane with a line y = 3/2x + 2 and a point (0, 2) b. coordinate plane with a line y = 3/2x 1 and a point (0, 1) c. coordinate plane with a line y = 1/2x + 2 and a point (2, 2) d. coordinate plane with a line y = 1/2x + 2 and a point (2, 3) e. coordinate plane with a line y = 2x + 2 and a point (0, 2) f. coordinate plane with a line y = 2x + 2 and a point (4, 0) In your own words, write the meaning of each vocabulary term. directed line segment a line connects two points in the shortest path line segment with points A and B labeled Theorems Theorem 3.13 Slopes of Parallel Lines In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope.   m₁ = m₂ Any two vertical lines are parallel. two vertical lines on a coordinate plane any two lines going straight up/down are parallel to each other. (since slope = N/A) Theorem 3.14 Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1.   Horizontal lines are perpendicular to vertical lines.   (m₁) (m₂) = -1 two perpendicular lines on a coordinate plane any line straight across with any line up/down will always be ⊥ y = mx + b Extra Practice 1. In Exercises 1 and 2, find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio.   1. A(2, 7), B(4, 1) (3 to 1) Find the slope: [\frac{7-1}{2-4} = \frac{6}{-2} = -3] Length of a unit: 4 units Divide the slope by the units: -3/4 [x = 2 + 3(\frac{3}{4}) = 2 + \frac{9}{4} = \frac{17}{4} = 4.25] [y = 7 + 3(\frac{-3}{4}) = 7 - \frac{9}{4} = \frac{19}{4} = 4.75] P(4.25, 4.75) 2. A(3, 1), B(8, -2) (2 to 3) Find the slope: [\frac{1-(-2)}{3-8} = \frac{3}{-5} = -\frac{3}{5}] Length from A to P is 2 units Length from B to P is 3 units Divide the slope by the units: -3/5 ÷ 2/5 = -3/2 [x = 3 + 2(\frac{-3}{2}) = 3 - 3 = 0] [y = 1 + 2(\frac{-3}{2}) = 1 - 3 = -2] P(0, -2) 3. Determine which of the lines are parallel and which of the lines are perpendicular. Find the slopes r: (2, 4), (-2, 1) [\frac{4-1}{2-(-2)} = \frac{3}{4}] s: (-1, 4), (1, -2) [\frac{4-(-2)}{-1-1} = \frac{6}{-2} = -3] t: (-1, 0), (1, 4) [\frac{0-4}{-1-1} = \frac{-4}{-2} = 2] p: (2, -4), (1, -1) [\frac{-4-(-1)}{2-1} = \frac{-3}{1} = -3] q: (0, 3), (-3, -1) [\frac{3-(-1)}{0-(-3)} = \frac{4}{3}] r || t s ⊥ p 4. Tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. Line 1: (2, 0), (-2, 2) [\frac{0-2}{2-(-2)} = \frac{-2}{4} = -\frac{1}{2}]   Line 2: (1, -2), (4, 4) [\frac{-2-4}{1-4} = \frac{-6}{-3} = 2] perpendicular Slopes are reciprocals 5. Write an equation of the line passing through point P(3, -2) that is parallel to y = 2/3x - 1. Graph the equations of the lines to check that they are parallel. Use point-slope formula. y - y₁ = m(x - x₁) y - (-2) = 2/3(x - 3) y + 2 = 2/3x - 2 y = 2/3x - 4 6. Write an equation of the line passing through point P(-2, 2) that is perpendicular to y = 2x + 3. Graph the equations of the lines to check that they are perpendicular. 1   1. Use y - y₁ = m(x - x₁) y - 2 = -1/2(x - (-2)) y - 2 = -1/2x - 1 y = -1/2x + 1 7. Find the distance from point A(0, 5) to y = -3x - 5 Find the equation of the perpendicular line that passes through A(0, 5) y = 1/3x + 5 Solve the system y = -3x - 5 y = 1/3x + 5 -3x - 5 = 1/3x + 5 -9x = x + 30 -10x = 30 x = -3 y = -3(-3) - 5 y = 9 - 5 y = 4 Use the Distance Formula √((0 - (-3))² + (5 - 4)²) √((0 + 3)² + (1)²) √(3)² + (1)² √9 + 1 √10