Parallel & Perpendicular Lines

Slope of Parallel Lines Two non-vertical lines with the same slope are parallel.

Parallel Lines The equations of two parallel lines have the same slope, but two different y-intercepts.

Parallel lines have the same slope If Line A has a slope of 4 (m = 4), then which of these four lines is parallel to Line A? Line B, m = 16 Line C, m = 12 Line D, m = 4 Line E, m = 2/3

y = 3x + 4 and y = 3x - 2 Y y = 3x + 4 X y = 3x -2 BACK

Are the two lines: L1, through (-2, 1) and (4, 5) L2, through (3, 0) and (0, -2), parallel?

Slopes of Perpendicular Lines If neither line is vertical, then the slopes of perpendicular lines are negative reciprocals. If the product of the slopes of two lines is -1 then the lines are perpendicular. Horizontal lines are perpendicular to vertical lines.

Perpendicular Slopes 4 3 What can we say about the intersection x-axis y-axis What can we say about the intersection of the two lines?

Write parallel perpendicular or neither for the pair of lines that passes through (5, -9) and (3, 7) and the line through (0, 2) and (8, 3).

Parallel lines have the same slope If Line A has a slope of 4 (m = 4), then which of these four lines is parallel to Line A? Line D, m = 4

Perpendicular lines have the ‘opposite reciprocal’ slope To get the opposite reciprocal… Express the slope as a fraction Then flip it and give it the opposite sign!

Perpendicular lines have the ‘opposite reciprocal’ slope So, a line with slope 3 is perpendicular to a line with slope –1/3

Perpendicular lines have the ‘opposite reciprocal’ slope If Line A has a slope of –4/5 (m = -4/5), then which of these four lines is perpendicular to Line A? Line B, m = 9 Line C, m = 5/4 Line D, m = 20 Line E, m = -5/4

Perpendicular lines have the ‘opposite reciprocal’ slope If Line A has a slope of –4/5 (m = -4/5), then which of these four lines is perpendicular to Line A? Line C, m = 5/4

Equations to Remember Slope-Intercept Form Useful for graphing since m is the slope and b is the y-intercept Point-Slope Form Use this form when you know a point on the line and the slope Also can use this version if you have two points on the line because you can first find the slope using the slope formula and then use one of the points and the slope in this equation. Standard Form Commonly used to write linear equation problems or express answers

Parallel and Perpendicular Remember parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same. Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.

Rules and Properties Parallel Lines If two different lines have the same slope, the lines are parallel. If two nonvertical lines are parallel, they have the same slope. Two parallel, vertical lines have undefined slopes.

Parallel lines have the same slope. Rules and Properties Parallel Lines Parallel lines have the same slope.

All vertical lines are parallel and have undefined slope. Rules and Properties All vertical lines are parallel and have undefined slope.

All horizontal lines are parallel and have a slope of 0. Rules and Properties All horizontal lines are parallel and have a slope of 0.

Find the Equation of a Parallel Line Write an equation in slope intercept form for the line that contains the point (-1, 3) and is parallel to the line y = -2x + 4. Lines that are parallel have the same slope. What is the slope of the given equation. m = -2 The values for y, m and x are given. Substitute these values and find the value of b.

Find the Equation of a Parallel Line The values for y, m and x are given. Substitute these values and find the value of b. Remember y = mx + b 3 = -2(-1) +b Multiply 3 = 2 + b Subtract 2 from both sides of the equal sign. 1 = b Write the equation by substituting for m and b. y = -2x + 1 This is parallel to y = -2x + 4

TRY THIS Find the Equation of a Parallel Line Write an equation in slope intercept form for the line that contains the point (-3, -4) and is parallel to the line y = -4x -2. Lines that are parallel have the same slope. What is the slope of the given equation. m = -4 The values for y, m and x are given. Substitute these values and find the value of b.

Find the Equation of a Parallel Line The values for y, m and x are given. Substitute these values and find the value of b. Remember y = mx + b -4 = -4(-3) +b Multiply -4 = 12 + b Subtract 12 from both sides of the equal sign. -16 = b Write the equation by substituting for m and b. y = -4x - 16 This is parallel to y = -4x -2

Rules and Properties Perpendicular Lines 5.6 Parallel and Perpendicular Lines Rules and Properties Perpendicular Lines If the slopes of two lines are m and , the lines are perpendicular. m 1 –

Rules and Properties Perpendicular Lines What angles are formed by perpendicular lines? Right angles which equal 900.

Rules and Properties Perpendicular Lines The slopes of perpendicular lines are opposite reciprocals of each other. All vertical and all horizontal lines are perpendicular to each other.

Find the Equation of a Perpendicular Line Write an equation in slope intercept form for the line that contains the point (-2, 3) and is perpendicular to the line y = -2x + 4. Lines that are perpendicular to each other have opposite reciprocal slopes. The slope for the original line is –2. What is the slope for the perpendicular line? 1/2

Find the Equation of a Perpendicular Line Remember y = mx + b The values for y, m and x are given. Substitute these values and find the value of b. y = 3, x = -2, m = (1/2) 3 = (1/2)(-2) +b Multiply. 3 = -1 + b Add 1 to both sides of the equal sign. 4 = b Write the equation by substituting. y = (1/2)x + 4. This is perpendicular to y = -2x + 4

TRY THIS Find the Equation of a Perpendicular Line Write an equation in slope intercept form for the line that contains the point (-1, 5) and is perpendicular to the line y = -4x - 2. Lines that are perpendicular to each other have opposite reciprocal slopes. The slope for the original line is –4. What is the slope for the perpendicular line? 1/4

Find the Equation of a Perpendicular Line Remember y = mx + b The values for y, m and x are given. Substitute these values and find the value of b. y = 5, x = -1, m = (1/4) 5 = (1/4)(-1) +b Multiply. 5 = (-1/4) + b Add (1/4) to both sides of the equal sign. 5 ¼ = b Write the equation by substituting. y = (1/4)x + 5¼. This is perpendicular to y = -4x - 2

Parallel & Perpendicular Lines Algebra 1 & Algebra 1 Honors