Slopes of parallel and perpendicular lines

About this lesson

Geometry › Coordinate geometry

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Slopes of parallel and perpendicular lines

Warm-up

Three quick questions from earlier lessons.

Discover

Rotate a line of slope 2 by 90∘90^\circ. Its "1 right, 2 up" becomes "2 left, 1 up". What is the new slope?

  1. −12-\tfrac{1}{2}
  2. 12\tfrac{1}{2}

Key idea

Slopes tell parallel and perpendicular apart.

Slope m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}. Parallel lines have equal slopes. Perpendicular lines (neither vertical) have slopes with product −1-1: m2=−1m1m_2 = -\dfrac{1}{m_1}.

Worked example

Line through (1,2)(1, 2) and (5,4)(5, 4).

m=12m = \tfrac{1}{2}
2/4.
Parallel: 12\tfrac{1}{2}
Same slope.
Perpendicular: −2-2
Negative reciprocal.

Your turn, guided

Your turn: perpendicular to slope 34\tfrac{3}{4}.

−43-\tfrac{4}{3}
Flip and change sign.

Practice ★

  1. A line has slope 1/81/8. Find the slope of a line perpendicular to it.
  2. A line has slope 8/48/4. Find the slope of a line perpendicular to it.
  1. Are the lines y=6x+1y = 6x + 1 and y=−16x+5y = -\tfrac{1}{ 6 }x + 5 parallel, perpendicular or neither?
    1. Perpendicular: the product of the slopes is −1
    2. Parallel
    3. Neither

Watch out!

Practice ★★

  1. Find the slope of the line through (0,5)(0, 5) and (−5,−3)(-5, -3).
  2. Find the slope of the line through (−5,−2)(-5, -2) and (5,4)(5, 4).

Exit ticket

Three questions to check you have got it.

  1. A line has slope 1/81/8. Find the slope of a line perpendicular to it.
  2. A line has slope 8/48/4. Find the slope of a line perpendicular to it.
  3. A line has slope 4/74/7. Find the slope of a line perpendicular to it.

Summary card

Parallel and perpendicular slopes

Parallel: equal. Perpendicular: m1m2=−1m_1 m_2 = -1.

Watch out: flip and change the sign.

End of the lesson

Next lesson: Equations of lines through a point Back to the chapter

Answers: Slopes of parallel and perpendicular lines

A

  1. −8-8
  2. −12- \frac{1}{2}

B

  1. Perpendicular: the product of the slopes is −1

C

  1. 85\frac{8}{5}
  2. 35\frac{3}{5}

D

  1. −8-8
  2. −12- \frac{1}{2}
  3. −74- \frac{7}{4}

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