Parallel and Perpendicular Lines

By

Parallel and perpendicular lines can be spotted by their slopes. Parallel lines have the same slope, so they never meet. Perpendicular lines cross at a right angle, and their slopes are opposite reciprocals. For example, a line with slope \tfrac{2}{3} is perpendicular to a line with slope -\tfrac{3}{2}.

Before this lesson: You should know how to find slope, use slope-intercept form, and write a line in point-slope form.

What Are Parallel and Perpendicular Lines?

Think of train tracks. The two rails run side by side and never touch. They are parallel. They have the same steepness, so they have the same slope.

Now think of the corner of a sheet of paper. The two edges meet at a perfect square corner, or a 90° right angle. Lines that meet like that are perpendicular.

For perpendicular lines, one line rises where the other falls. Their slopes are opposite reciprocals. To get one from the other, flip the fraction and change the sign.

Two graphs side by side on grids from −5 to 5. Left, parallel lines: y = 1/2 x + 2 in blue and y = 1/2 x − 2 in orange, running side by side with the same slope and never meeting. Right, perpendicular lines: y = 2x − 1 in blue and y = −1/2 x + 4 in orange, crossing at (2, 3) with a small square marking the right angle.
Parallel lines have the same slope. Perpendicular lines have slopes like 2 and −1/2 that multiply to −1.

Slopes of Parallel and Perpendicular Lines

LinesSlope ruleExample
ParallelSame slope: m_1 = m_2, but different y-interceptsy = 3x + 1 and y = 3x - 4
PerpendicularOpposite reciprocals: m_2 = -\tfrac{1}{m_1}, so m_1 \cdot m_2 = -1y = 2x - 1 and y = -\tfrac{1}{2}x + 4
NeitherSlopes are different and don’t multiply to −1y = 4x and y = -4x

Here is how to find an opposite reciprocal. Start with \tfrac{2}{3}. Flip it to get \tfrac{3}{2}. Change the sign to get -\tfrac{3}{2}. Check: \tfrac{2}{3} \cdot \left(-\tfrac{3}{2}\right) = -1.

A whole number works the same way. Write 4 as \tfrac{4}{1}. Its opposite reciprocal is -\tfrac{1}{4}.

Steps to write a parallel or perpendicular line

  1. Find the slope of the given line. If it isn’t in y = mx + b form, solve for y first.
  2. For a parallel line, use the same slope. For a perpendicular line, use the opposite reciprocal.
  3. Put the new slope and the given point into y - y_1 = m(x - x_1).
  4. Solve for y and check that the point works.

Worked Examples

Example 1: Parallel, perpendicular or neither?

  • y = 3x + 1 and y = 3x - 4: both slopes are 3, and the y-intercepts are different. The lines are parallel.
  • y = \tfrac{2}{5}x and y = -\tfrac{5}{2}x + 1: \tfrac{2}{5} \cdot \left(-\tfrac{5}{2}\right) = -1. The lines are perpendicular.
  • y = 4x and y = -4x: the slopes are different, and 4 \cdot (-4) = -16, not -1. The lines are neither.

Watch the last pair. The signs are opposite, but 4 was not flipped. So the lines are not perpendicular.

Example 2: Lines in standard form

Are 2x + 3y = 6 and 4x + 6y = -12 parallel, perpendicular or neither?

Solve each one for y.

\begin{aligned} 2x + 3y &= 6 &\quad 4x + 6y &= -12 \\ 3y &= -2x + 6 &\quad 6y &= -4x - 12 \\ y &= -\tfrac{2}{3}x + 2 &\quad y &= -\tfrac{2}{3}x - 2 \end{aligned}

Answer: Both slopes are -\tfrac{2}{3}, and the y-intercepts are 2 and -2. The lines are parallel.

Example 3: Write a parallel line

Write the equation of the line parallel to y = -2x + 5 that passes through (3, 1).

A parallel line has the same slope, so m = -2. Use point-slope form.

\begin{aligned} y - 1 &= -2(x - 3) \\ y - 1 &= -2x + 6 \\ y &= -2x + 7 \end{aligned}

Answer: y = -2x + 7. Check: -2(3) + 7 = 1. It works.

Example 4: Write a perpendicular line

Write the equation of the line perpendicular to y = \tfrac{2}{3}x - 1 that passes through (4, -1).

The given slope is \tfrac{2}{3}. Flip it and change the sign: m = -\tfrac{3}{2}.

\begin{aligned} y - (-1) &= -\tfrac{3}{2}(x - 4) \\ y + 1 &= -\tfrac{3}{2}x + 6 \\ y &= -\tfrac{3}{2}x + 5 \end{aligned}

Answer: y = -\tfrac{3}{2}x + 5. Check: -\tfrac{3}{2}(4) + 5 = -6 + 5 = -1. It works.

Example 5: Word problem with a street map

On a city map, Main Street passes through (0, 2) and (4, 4). The city plans a new road that crosses Main Street at (4, 4) at a right angle. Write the equation of the new road.

First find the slope of Main Street: m = \tfrac{4 - 2}{4 - 0} = \tfrac{2}{4} = \tfrac{1}{2}.

The new road is perpendicular, so its slope is the opposite reciprocal: -2. It passes through (4, 4).

\begin{aligned} y - 4 &= -2(x - 4) \\ y - 4 &= -2x + 8 \\ y &= -2x + 12 \end{aligned}
A street map on a grid with x and y from −1 to 9. Main Street, in blue, passes through (0, 2) and (4, 4) and has slope 1/2. A new road, in orange, crosses Main Street at (4, 4) at a right angle. It has slope −2 and the equation y = −2x + 12. A small square marks the right angle.
The new road’s slope, −2, is the opposite reciprocal of Main Street’s slope, 1/2.

Answer: The new road is y = -2x + 12.

Common Mistakes

MistakeExample of the errorFix
Flipping but not changing the signPerpendicular to slope \tfrac{2}{3}: using \tfrac{3}{2}Flip and change the sign: -\tfrac{3}{2}.
Changing the sign but not flippingPerpendicular to slope 4: using -4Flip too: -\tfrac{1}{4}.
Reading slope before solving for ySaying 2x + 3y = 6 has slope 2Solve for y first: the slope is -\tfrac{2}{3}.
Calling the same line parallely = 2x + 1 and 2y = 4x + 2They are the same line. Parallel lines need different y-intercepts.

Special Cases and Other Uses

  • Horizontal and vertical lines are always perpendicular. For example, y = 3 and x = -1 meet at a right angle. You can’t multiply their slopes, because a vertical line has no slope. See horizontal and vertical lines.
  • Two horizontal lines, like y = 1 and y = 5, are parallel. So are two vertical lines.
  • Standard form shortcut: lines in standard form with the same A and B, like 3x - y = 4 and 3x - y = 7, are parallel.
  • Systems: two parallel lines never cross, so their system has no solution. See number of solutions for systems of equations.
  • Right angles in shapes: if two sides of a triangle have slopes that multiply to −1, the triangle has a right angle.

Practice Problems

For 1–3, decide if the lines are parallel, perpendicular or neither.

  1. y = 5x - 2 and y = 5x + 3
  2. y = -\tfrac{3}{4}x + 1 and y = \tfrac{4}{3}x - 2
  3. y = 2x + 1 and y = \tfrac{1}{2}x + 1
  4. Find the slope of a line parallel to y = -6x + 1 and the slope of a line perpendicular to it.
  5. Are 3x - y = 4 and 6x - 2y = 10 parallel, perpendicular or neither?
  6. Write the line parallel to y = 4x - 3 through (1, 6).
  7. Write the line perpendicular to y = -3x + 2 through (6, 1).
  8. Write the line perpendicular to 2x + 5y = 10 through (2, 3).
  9. Write the line perpendicular to y = 4 through (-3, 2).
  10. A triangle has corners P(0, 0), Q(3, 1) and R(2, 4). Is angle Q a right angle?
Show answers
  1. Parallel. Both slopes are 5, and the y-intercepts are different.
  2. Perpendicular. -\tfrac{3}{4} \cdot \tfrac{4}{3} = -1.
  3. Neither. The slopes are 2 and \tfrac{1}{2}. They are reciprocals, but the sign did not change, so the product is 1, not -1.
  4. Parallel slope: -6. Perpendicular slope: \tfrac{1}{6}.
  5. Parallel. 3x - y = 4 becomes y = 3x - 4. 6x - 2y = 10 becomes y = 3x - 5. Same slope, different y-intercepts.
  6. y - 6 = 4(x - 1), so y = 4x + 2.
  7. The perpendicular slope is \tfrac{1}{3}. y - 1 = \tfrac{1}{3}(x - 6), so y = \tfrac{1}{3}x - 1.
  8. 2x + 5y = 10 has slope -\tfrac{2}{5}, so the perpendicular slope is \tfrac{5}{2}. y - 3 = \tfrac{5}{2}(x - 2), so y = \tfrac{5}{2}x - 2.
  9. y = 4 is horizontal, so the perpendicular line is vertical. Through (-3, 2), it is x = -3.
  10. Slope of PQ: \tfrac{1 - 0}{3 - 0} = \tfrac{1}{3}. Slope of QR: \tfrac{4 - 1}{2 - 3} = -3. Since \tfrac{1}{3} \cdot (-3) = -1, the sides are perpendicular. Yes, angle Q is a right angle.

Parallel and Perpendicular Lines FAQ

How do you know if two lines are parallel?

Write both in y = mx + b form. If the slopes are equal and the y-intercepts are different, the lines are parallel.

How do you find the slope of a perpendicular line?

Take the opposite reciprocal: flip the fraction and change the sign. The slope \tfrac{3}{5} becomes -\tfrac{5}{3}, and the two slopes multiply to -1.

Are horizontal and vertical lines perpendicular?

Yes. They always meet at a right angle. The slope rule doesn’t apply, because a vertical line has no slope.

Do perpendicular lines have to cross at the origin?

No. They can cross at any point. In Example 4, the lines cross at (4, -1).

Summary

  • Parallel lines have the same slope and different y-intercepts. They never meet.
  • Perpendicular lines meet at a right angle. Their slopes are opposite reciprocals that multiply to -1.
  • To find an opposite reciprocal, flip the fraction and change the sign.
  • To write a parallel or perpendicular line, find the new slope, then use point-slope form with the given point.
  • Horizontal and vertical lines are perpendicular to each other.

Previous lesson: Standard Form of a Linear Equation

Next lesson: Direct Variation

Back to the Algebra 1 Index

Similar Posts