Perpendicular Gradients

\( m_1\,m_2=-1 \)
Coordinate Geometry GCSE
Question 2 of 20
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Check if slopes 2 and -1/2 are perpendicular.

Hint (H)
\( Multiply slopes, check if product=-1. \)

Explanation

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Statement

If two lines are perpendicular, their gradients multiply to give \(-1\):

\[ m_1 m_2 = -1 \]

Why it’s true

  • The slope of a line measures its steepness.
  • If two lines are perpendicular, one is the negative reciprocal of the other.
  • That is, if one slope is \(m_1\), then the other must be \(m_2 = -1/m_1\).
  • Multiplying them gives \(m_1 m_2 = -1\).

Recipe (how to use it)

  1. Find slope of one line (\(m_1\)).
  2. Use formula \(m_2 = -1/m_1\) for the perpendicular line.
  3. If both slopes are known, check if their product is \(-1\).

Spotting it

This formula is used when checking whether two lines are perpendicular, or when finding the slope of a perpendicular line through a given point.

Common pairings

  • Slope-intercept form of a line: \(y=mx+c\).
  • Midpoint and perpendicular bisectors.
  • Circle tangent and radius problems.

Mini examples

  1. Given: Line slope = 2. Perpendicular slope: \(-1/2\).
  2. Given: One line slope = -3, other line slope = 1/3. Product = -1, so they are perpendicular.

Pitfalls

  • Forgetting to take the negative reciprocal.
  • Confusing perpendicular with parallel (\(m_1 = m_2\)).

Exam strategy

  • Check product of slopes quickly to confirm perpendicularity.
  • Special case: vertical line (\(m=\infty\)) and horizontal line (\(m=0\)) are perpendicular.

Summary

Two lines are perpendicular if their gradients multiply to give \(-1\). Use \(m_2=-1/m_1\) to find the slope of a perpendicular line.

Worked examples

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  1. \( Line slope=2. Find slope of perpendicular line. \)
    1. \( m2=-1/m1=-1/2 \)
    Answer: -1/2
  2. \( Line slope=3. Find slope of perpendicular line. \)
    1. \( m2=-1/3 \)
    Answer: -1/3
  3. \( Line slope=-4. Find slope of perpendicular line. \)
    1. \( m2=-1/(-4)=1/4 \)
    Answer: 1/4
  4. \( Line slope=1/2. Find slope of perpendicular line. \)
    1. \( m2=-1/(1/2)=-2 \)
    Answer: -2
  5. \( Line slope=1/3. Find slope of perpendicular line. \)
    1. \( m2=-1/(1/3)=-3 \)
    Answer: -3
  6. \( Line slope=5. Find slope of perpendicular line. \)
    1. \( m2=-1/5 \)
    Answer: -1/5
  7. Check if slopes 2 and -1/2 are perpendicular.
    1. \( 2*(-1/2)=-1, so yes \)
    Answer: Yes
  8. Check if slopes 3 and 1/3 are perpendicular.
    1. \( 3*(1/3)=1, not -1 \)
    Answer: No
  9. Check if slopes -3 and 1/3 are perpendicular.
    1. \( -3*(1/3)=-1, so yes \)
    Answer: Yes
  10. Check if slopes 0 and undefined line are perpendicular.
    1. Horizontal and vertical lines are perpendicular.
    Answer: Yes