How to find parallel and perpendicular slopes?

How to find parallel and perpendicular slopes from an equation

Every non-vertical linear equation can be rearranged into slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. For example, if a line has the equation 2x + 4y = 8, solve for y by subtracting 2x and dividing every term by 4. This yields y = -1/2 x + 2, showing that the slope m is -1/2.

Once you isolate the slope m of the reference line, you can instantly determine the slopes of any lines that run parallel or perpendicular to it.

Parallel slopes are identical

Parallel lines never intersect on the coordinate plane because they tilt at the exact same angle. Consequently, their slopes are always equal (m1 = m2).

If your original line has a slope of 3, any line parallel to it also has a slope of 3. If you need the equation of a parallel line passing through a specific point like (2, 5), use the point-slope form: y - 5 = 3(x - 2).

Perpendicular slopes are negative reciprocals

Perpendicular lines intersect at a right angle (90 degrees). Their slopes are negative reciprocals, which means you flip the fraction upside down and invert the sign. Mathematically, m1 * m2 = -1, or m2 = -1 / m1.

For example, if a line has a slope of 2/5, the perpendicular slope is -5/2. If the initial slope is -4 (which is -4/1), the perpendicular slope is +1/4. The only exception involves horizontal and vertical lines: a horizontal line has a slope of 0, and a perpendicular vertical line has an undefined slope.

Writing equations and finding the perpendicular bisector

A common application in 11th grade geometry is finding the perpendicular bisector of a segment AB. First, calculate the midpoint of segment AB using the midpoint formula. Next, find the slope of segment AB using (y2 - y1) / (x2 - x1). Then, take the negative reciprocal of that slope. Finally, write the line equation using the perpendicular slope and the midpoint coordinates.