Year 10 Standard · Unit 2 Generalizing and coordinate geometry · Lesson 4 of 6

How steep, and which way is square on?

Gradient measures steepness as rise over run. Lines with the same gradient are parallel, and lines at right angles have gradients that multiply to negative one. These are rules for every line.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: write 3/8 as a decimal.
    Answer0.375.
  2. From Year 8: what is the reciprocal of 10?
    Answer1/10.
  3. From Year 9: what is the gradient of the line y = 4x − 7?
    Answer4, the number in front of x.
Words used in this lesson
gradient
how steep a line is: rise divided by run. For example: up 3 for 5 across.
rise
the change in the y value. For example: up.
run
the change in the x value. For example: across.
parallel
lines with the same gradient that never meet. For example: railway tracks.
perpendicular
lines at a right angle to each other. For example: corner of a page.
reciprocal
one divided by the number. For example: 1/7 for 7.

Which ramp is steeper?

A school in Singapore plans two ramps. Ramp A rises 1 m over 4 m of horizontal distance. Ramp B rises 2 m over 9 m. Ramp B rises more, but it is also longer.

Predict first: which ramp is steeper, or are they the same? Write down your choice and a reason, then use the first figure to make a rise and run of your own and read the gradient.

−4
−2
1
1
3/5gradient from A to B
60%the same gradient as a percentage

A(−4, −2) and B(1, 1): rise 3, run 5, so the gradient is 3 ÷ 5 = 3/5 = 0.6, which is 60%.

Show me the steps
  1. A(−4, −2) and B(1, 1): rise 3, run 5. The gradient is 3/5 = 0.6, which is 60%.
  2. A(−4, 3) and B(2, −3): rise −6, run 6. The gradient is −1: the line goes down as it goes right.
  3. A(−3, 1) and B(4, 1): rise 0. The line is horizontal and its gradient is 0.
  4. A(2, −4) and B(2, 3): run 0. The line is vertical, and dividing by zero is not possible, so it has no gradient.

Gradient = rise ÷ run, the change in y over the change in x. It can be written as a fraction, a decimal or a percentage: 3/5 = 0.6 = 60%. A horizontal line has gradient 0, and a vertical line has no gradient, because its run is 0.

Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1: turn the gradient upside down and change its sign.

1
3
1/3gradient of the first line
−3gradient of the perpendicular line

Rise 1, run 3: the first line has gradient 1/3. Turned through a right angle the rise and run become run 1 and rise −3, so the perpendicular gradient is −3 and 1/3 × (−3) = −1.

Show me the steps
  1. Rise 1, run 3: gradient 1/3. The perpendicular line has gradient −3, and 1/3 × (−3) = −1.
  2. Rise 3, run 1: gradient 3. The perpendicular line has gradient −1/3, and 3 × (−1/3) = −1.
  3. Rise 0, run 2: the first line is horizontal (gradient 0). Its perpendicular is vertical, which has no gradient, so the product rule does not apply.
  4. Rise −2, run 2: gradient −1. The perpendicular line has gradient 1, and −1 × 1 = −1.

Two common mistakes. Writing run over rise: a line that rises 2 over 5 has gradient 2/5, not 5/2. And changing only the sign for a perpendicular line, or only flipping: the perpendicular to 2/3 is −3/2, with both the flip and the change of sign.

Worked examples

1. Find the gradient of the line through (−2, 5) and (3, −5).

  1. Rise: −5 − 5 = −10. Run: 3 − (−2) = 5.
  2. Gradient = −10 ÷ 5 = −2. The line falls as it goes right.

2. Are the line through (0, 1) and (4, 9) and the line through (1, −3) and (3, 1) parallel?

  1. First: (9 − 1) ÷ (4 − 0) = 8 ÷ 4 = 2. Second: (1 − (−3)) ÷ (3 − 1) = 4 ÷ 2 = 2.
  2. The gradients are equal, so the lines are parallel.

3. A line has gradient 2/3. Find the gradient of a line perpendicular to it.

  1. Turn it upside down: 3/2. Change the sign: −3/2.
  2. Check: 2/3 × (−3/2) = −1, so the perpendicular gradient is −3/2.

Your go. The first step is given. A line has gradient −5. Find the gradient of a line perpendicular to it. Write −5 as −5/1.

  1. −5 = −5/1. Turn it upside down: 1/(−5).
  2. Turning over keeps the minus sign, so the sign still has to change. Do that, then check that the product with −5 is −1.
Check the answer

Upside down gives 1/(−5) = −1/5, and the sign must change, so the perpendicular gradient is 1/5. Check: −5 × 1/5 = −1.

Check before you go on

1. A line has gradient 5/2. What is the gradient of a line perpendicular to it?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Find the gradient of the line through (1, 2) and (4, 14).

3. A mountain road in Bhutan climbs 18 m over a horizontal distance of 150 m. What is its gradient as a percentage?

Level 2 · Building

4. Find the gradient of the line through (−2, 7) and (2, −3) as a decimal.

5. A line has gradient 0.8. What is the gradient of a line perpendicular to it, as a decimal?

6. The line through (0, 1) and (2, k) is parallel to the line through (0, 3) and (4, 11). What is k?

Level 3 · Stretch

7. The line through (1, 2) and (5, 3) is perpendicular to the line through the origin (0, 0) and (6, k). What is k?

8. A is (0, 0), B is (6, 2) and C is (−2, k). The angle at A is a right angle. What is k?

Something to think about

Two lines are perpendicular and both have positive gradients. Is that possible? Use m₁ × m₂ = −1 to decide.

One way to think about it

No. A product of −1 is negative, so one gradient is positive and the other negative. On the grid: one line rises as it goes right and a line at a right angle to it must fall as it goes right. The exception is a horizontal and a vertical line, where one gradient is 0 and the other does not exist.

Exit ticket

9. Line L passes through (1, 1) and (4, 7). Which of these lines, each given by two points, is perpendicular to L?