4.3 Writing equations of parallel and perpendicular lines Today’s learning goal is that students will be able to identify and write equations of parallel and perpendicular lines and use them in real-life problems.

Core Concept Parallel lines lie in the same plane and never intersect. have the same slope. All vertical lines are parallel. Perpendicular lines Lie in the same plane and their intersection creates a right angle. Their slopes are opposite reciprocals. Vertical lines are perpendicular to horizontal lines

Example 1 – identifying parallel lines Determine which of the lines are parallel Line a: 𝑚= 2 −3 1 −(−4) = −1 5 Line b: 𝑚= −1 −0 1−(−3) = −1 4 Line c: 𝑚= −5 −(−4) 2−(−3) = −1 5 Lines a and c are parallel They both have a slope of −1 5

Example 2 – Writing an Equation of a parallel line Write an equation of the line that passes through (5, -4) and is parallel to the line y = 2x+3. Step 1 – Find the slope of the parallel line. The line has a slope of 2, the parallel line will also have a slope of 2. Step 2 – Use the slope-intercept form to find the y-intercept of the parallel line. y = mx + b Write the slope-intercept form -4 = 2 (5) + b Substitute -4 for y, 5 for x and 2 for m. -4 = 10 + b Simplify -10 -10 Solve for b. -14 = b Use m = 2 and b = -14 to write the equation y = 2x – 14

You Try! 1. Line a passes through (-5, 3) and (-6, -1). Line b passes through (3, -2) and (2, -7). Are the lines parallel? Explain. 2. Write an equation of the line that passes through (-4, 2) and is parallel to the line y = ¼ x +1 . No, they do not have the same slope. 2 = ¼ (-4) + b 2 = -1 + b 3 = b y = ¼ x + 3

Example 3 – identifying parallel and perpendicular lines Determine which of the lines, if any, are parallel or perpendicular. Line a: y = 4x +2 y = 4x + 2 Slope = 4 Line b: x + 4y = 3 -x -x 4y = -x + 3 4 4 4 y = - ¼ x + ¾ Slope = - ¼ Line c: -8y – 2x = 16 +2x +2x -8y = 2x + 16 -8 -8 -8 y = - ¼ x – 2 Slope = - ¼ Write the equations in slope-intercept form. Then compare the slopes. 1 1 Lines b and c are parallel. Line a is perpendicular to lines b and c.

Example 4 – Writing an Equation of a perpendicular line Write an equation of the line that passes through (-3, 1) and is perpendicular to the line y = ½x + 3. Step 1 – Find the slope of the line then change it to the opposite reciprocal. The line has a slope of 1/2, the perpendicular line has a slope of -2. Step 2 – Use the slope m = -2 and the point (-3, 1) with the point-slope form. 𝑦 −𝑦1=𝑚 (𝑥 −𝑥1) Write the point-slope form. 𝑦 −1=−2(𝑥 −(−3)) Substitute -2 for m, -3 for x1, and, 1 for y1 𝑦 −1=−2(𝑥+3) Simplify 𝑦 −1=−2𝑥 −6 Distribute +1 +1 Solve for y. 𝑦 =−2𝑥−5 x1, y1

You Try! 3. Determine which of the lines, if any, are parallel, or perpendicular. Line a: 2x + 6y = -3 Line b: y = 3x – 8 Line c: -6y + 18x = 9 4. Write an equation of the line that passes through (-3, 5) and is perpendicular to the line y = -3x -1, Lines b and c are parallel. Line a is perpendicular to lines b and c. y = 1/3 x + 6

Example 5 – Writing an equation of a perpendicular line The position of a helicopter search and rescue crew is shown in the graph. The shortest flight path to the shoreline is one that is perpendicular to the shoreline. Write an equation that represents this path.

Understand the Problem – You are asked to write an equation that represents the shortest path from the helicopter to the shoreline. (perpendicular distance) Make a plan – to write an equation for a perpendicular line you must: find the slope of the shoreline, find the opposite reciprocal slope. Then use point-slope form with the coordinate (14,4) to write the equation.

Complete the plan – The slope of the shoreline passes through the points (1, 3) and (4, 1) so, the slope is: 𝑚= 1−3 4−1 = −2 3 The opposite reciprocal slope is 3 2 .

Use the slope m = 3 2 and (14, 4) with the point-slope form. Complete the plan – Use the slope m = 3 2 and (14, 4) with the point-slope form. 𝑦 −𝑦1=𝑚 (𝑥 −𝑥1) Write the point-slope form. 𝑦 −4= 3 2 (𝑥 −14) Substitute. 𝑦 −4= 3 2 𝑥 − 42 2 Distribute. 𝑦 −4= 3 2 𝑥 −21 Simplify. +4 +4 Solve for y. 𝑦 = 3 2 𝑥 −17 Slope-intercept form. x1, y1

You try! 5. In Example 5, a boat is traveling parallel to the shoreline and passes through (9, 3). Write an equation that represents the path of the boat. boat (9,3) 𝑦= −2 3 𝑥+9