Perpendicular line equations

Two lines are perpendicular if they intersect at a right angle (90 degrees).

The key thing to remember when looking at their equations is that:

the gradients of perpendicular lines are negative reciprocals of each other

Finding the negative reciprocal

To find the negative reciprocal of a number, you:

• Take the reciprocal of the number (flip its fraction form upside down)
• Change its sign (make it negative if it was positive, or positive if it was negative)

If you’re using a calculator, just type in -1 \div m to find the negative reciprocal of a gradient m.

Example: find the negative reciprocal of 2

• 2 can be written as the fraction \frac{2}{1}
• The reciprocal of \frac{2}{1} is \frac{1}{2}
• Changing the sign gives -\frac{1}{2}
• Answer: -\frac{1}{2}

Example: find the negative reciprocal of -\frac{3}{4}

• The reciprocal of -\frac{3}{4} is -\frac{4}{3}
• Changing the sign gives \frac{4}{3}
• Answer: \frac{4}{3}

Finding the equation of a perpendicular line

Once you know the gradient of the new line, the steps are exactly the same as they are for parallel line equations:

1. Find the negative reciprocal of the gradient of the original line to get the gradient of the new line.
2. Substitute the gradient and the coordinates of the given point into the straight line equation: y = mx + c.
3. Solve for c to find the y intercept of the new line.
4. Write the equation of the new line using the gradient and y intercept.

Example: find the equation of the line perpendicular to y = 2x + 3 that goes through the point (4, 5)

• Gradient of original line is 2

• Negative reciprocal of 2 is -\frac{1}{2}

• So the gradient of the new line is -\frac{1}{2}

• We know that one of the points on the new line is (4, 5), so when x=4, y=5.

• We can substitute these values into the equation of a straight line:

  • y = mx + c
  • 5 = -\frac{1}{2} \times 4 + c
  • 5 = -2 + c
  • c = 5 + 2
  • c = 7

• The y-intercept (c) is 7, so write the equation for a line with gradient -\frac{1}{2} and y-intercept 7:

  • y = -\frac{1}{2}x + 7

• Answer: y = -\frac{1}{2}x + 7

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