Perpendicular Lines – Meeting at Right Angles
Two lines are perpendicular if they cross at a perfect 90° right angle. The gradient test for this is one of the more surprising rules in coordinate geometry: two lines are perpendicular exactly when the product of their gradients equals −1. Multiply the two gradients together, and if you get exactly −1, the lines meet at a right angle.
A quick shortcut follows directly from this rule: to find the gradient perpendicular to a given gradient m, flip it upside down and change its sign – the perpendicular gradient is always −1/m.
Checking for Perpendicular Lines
Lines are perpendicular when m₁ × m₂ = −1. Given a gradient m, the perpendicular gradient is −1/m.
Gradient of A = (4 − 0) ÷ (2 − 0) = 2. Gradient of B = (3 − 5) ÷ (5 − 1) = −2 ÷ 4 = −0.5. Product: 2 × (−0.5) = −1, so the lines are perpendicular.
Flip and negate: −1 ÷ 4 = −1/4.
Real-Life Application
- Building corners: walls meeting at right angles are perpendicular by design.
- Road intersections: many city streets cross at perpendicular right angles.
- Sports courts: boundary lines on courts and pitches are laid out perpendicular to each other.
Key Takeaways
- Perpendicular lines meet at a right angle, and their gradients multiply to −1.
- The perpendicular gradient of m is always −1/m.
- This gradient test works from any two points on each line.
Practice: Perpendicular Lines
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