Coordinate Geometry and Conics · Lesson 2

Parallel and perpendicular lines

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1-1: flip the fraction and change the sign. A line perpendicular to one with slope 23\frac{2}{3} has slope −32-\frac{3}{2}. A horizontal line and a vertical line are also perpendicular.

Standard form. The line Ax+By=CAx + By = C has slope −AB-\frac{A}{B}. You can also solve for y: 3x−4y=123x - 4y = 12 becomes y=34x−3y = \frac{3}{4}x - 3, so its slope is 34\frac{3}{4}.

A line through a point. With slope m and a point (x1,y1)(x_1, y_1), write y−y1=m(x−x1)y - y_1 = m(x - x_1) and simplify. The perpendicular bisector of a segment passes through the segment's midpoint at a right angle: find the midpoint, find the segment's slope, take its negative reciprocal, then use point-slope form.

Worked example Easy

In the standard (x, y) coordinate plane, line ℓ\ell has equation 3x−4y=123x - 4y = 12. What is the slope of a line perpendicular to ℓ\ell?

  1. −43-\frac{4}{3}Answer
  2. −34-\frac{3}{4}
  3. 34\frac{3}{4}
  4. 43\frac{4}{3}

How to solve it

  1. Slope of ℓ\ell: −AB=−3−4=34-\frac{A}{B} = -\frac{3}{-4} = \frac{3}{4}.
  2. Perpendicular slope: flip the fraction and change the sign, −43-\frac{4}{3}.
  3. Check: 34⋅(−43)=−1\frac{3}{4} \cdot \left(-\frac{4}{3}\right) = -1.

Why each choice is right or wrong

  • A. Correct. Line ℓ\ell has slope −3−4=34-\frac{3}{-4} = \frac{3}{4}, and its negative reciprocal is −43-\frac{4}{3}.
  • B. Incorrect. −34-\frac{3}{4} changes the sign of the slope of ℓ\ell but doesn't flip the fraction. Its product with 34\frac{3}{4} is −916-\frac{9}{16}, not −1-1.
  • C. Incorrect. 34\frac{3}{4} is the slope of ℓ\ell itself, which gives a parallel line, not a perpendicular one.
  • D. Incorrect. 43\frac{4}{3} flips the fraction but keeps the positive sign. Perpendicular slopes have opposite signs.

Worked example Medium

Which equation describes the line in the standard (x, y) coordinate plane that passes through (6,2)(6, 2) and is perpendicular to the line y=−23x+1y = -\frac{2}{3}x + 1?

  1. y=−32x+11y = -\frac{3}{2}x + 11
  2. y=−23x+6y = -\frac{2}{3}x + 6
  3. y=23x−2y = \frac{2}{3}x - 2
  4. y=32x−7y = \frac{3}{2}x - 7Answer

How to solve it

  1. The given slope is −23-\frac{2}{3}, so the perpendicular slope is 32\frac{3}{2}.
  2. Point-slope form: y−2=32(x−6)y - 2 = \frac{3}{2}(x - 6).
  3. Simplify: y=32x−9+2=32x−7y = \frac{3}{2}x - 9 + 2 = \frac{3}{2}x - 7.

Why each choice is right or wrong

  • A. Incorrect. y=−32x+11y = -\frac{3}{2}x + 11 flips the fraction but keeps the negative sign, so its slope times −23-\frac{2}{3} is 1, not −1-1.
  • B. Incorrect. y=−23x+6y = -\frac{2}{3}x + 6 keeps the original slope, so it is the parallel line through (6,2)(6, 2).
  • C. Incorrect. y=23x−2y = \frac{2}{3}x - 2 changes the sign without flipping the fraction. Its slope times −23-\frac{2}{3} is −49-\frac{4}{9}.
  • D. Correct. The perpendicular slope is 32\frac{3}{2}, and y−2=32(x−6)y - 2 = \frac{3}{2}(x - 6) simplifies to y=32x−7y = \frac{3}{2}x - 7. Check the point: 32(6)−7=2\frac{3}{2}(6) - 7 = 2.

Trap Negative, reciprocal, or both

A perpendicular slope needs both moves. From 23\frac{2}{3}, check that your slope has both changes: −23-\frac{2}{3} only changes the sign, 32\frac{3}{2} only flips the fraction, and 23\frac{2}{3} is the parallel slope. The perpendicular slope is −32-\frac{3}{2}.

Calculator When a calculator isn’t faster

Finding a slope and writing a line by hand takes seconds, so a calculator rarely helps here. If you test online and want a check, graph the given line and your answer in the built-in Desmos calculator: they should cross at your point and look square to each other.