Year 9 · Unit 3 Linear relationships · Lesson 6 of 7

When do two lines never meet, and when do they cross squarely?

Compare two gradients and decide whether the lines are parallel, perpendicular or neither, then write the equation of a line through a point that is parallel or perpendicular to a given line.

FunctionsNo calculator

Warm-up

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

  1. (Lesson 3) Write down the gradient of y = −3x + 7.
    Answer−3.
  2. (Lesson 4) A line with gradient 2 passes through (1, 5). What is c?
    Answer3: 5 = 2 × 1 + c.
  3. What is the reciprocal of 3/4?
    Answer4/3: flip the fraction.
Words used in this lesson
parallel
never meeting, with equal gradients. For example: y = 2x + 1 and y = 2x − 4.
perpendicular
meeting at a right angle. For example: gradients 2 and −1/2.
reciprocal
a number flipped: 1 divided by it. For example: the reciprocal of 4 is 1/4.
negative reciprocal
the reciprocal with the opposite sign. For example: the negative reciprocal of 2 is −1/2.
right angle
an angle of 90 degrees. For example: the corner of a square.
product
the answer to a multiplication. For example: the product of 2 and −1/2 is −1.

A road and a path

A town planner draws a main road on a map along y = 2x. A cycle lane is to run beside it, never meeting it. A footpath is to cross it at a right angle. What must be true of the gradient of the lane, and of the footpath?

Predict first: the lane's gradient is quick to guess. For the footpath, many people say “1/2”, a flipped 2. Is that right? Try gradients in the first figure: start where it opens, and try to find the footpath.

2
1/2
2first gradient
1/2second gradient
1product
neitherthe lines are

Gradients 2 and 1/2: the product is 1. The lines are neither parallel nor perpendicular: the product is 1, which is not −1, and the gradients are not equal.

Show me the steps
  1. 2 and 1/2: the product is 1, not −1. They are neither: a reciprocal alone is not enough.
  2. 2 and 2: equal gradients. Parallel: the lines never meet.
  3. 2/3 and −3/2: the product is −1. Perpendicular.
  4. −3 and 1/3: the product is −1. Perpendicular.
  5. 1 and −1: the product is −1. Perpendicular, each at 45 degrees to the axes.

Parallel lines have equal gradients and different values of c, so they never meet. Perpendicular lines have gradients that multiply to −1: m1 × m2 = −1, so m2 = −1 ÷ m1: flip the fraction and change the sign.

The footpath for y = 2x needs gradient −1/2. Check: 2 × (−1/2) = −1.

One exception: a horizontal line and a vertical line are perpendicular, but the rule cannot show it: gradient 0 and no gradient.

A common mistake: only flipping, or only changing the sign. For gradient 2, the flip gives 1/2 and the sign change gives −2. Neither multiplies with 2 to give −1: 2 × 1/2 = 1 and 2 × (−2) = −4. You need both.

The perpendicular through a point

To write the equation of the perpendicular through a given point, take the negative reciprocal of the gradient, then find c by putting the point in, as in Lesson 4. The small square marks the right angle.

1
−2new gradient
7new c
y = −2x + 7perpendicular line

Given y = x/2 + 1 and the point (2, 3): the new gradient is −2. Putting the point in gives c = 7. The line is y = −2x + 7.

Show me the steps
  1. y = x/2 + 1 through (2, 3): new gradient −2. 3 = −2 × 2 + c, so c = 7: y = −2x + 7.
  2. y = −3x + 2 through (3, −1): new gradient 1/3. −1 = 1/3 × 3 + c, so c = −2: y = x/3 − 2.
  3. y = 2x/3 − 1 through (−2, 1): new gradient −3/2. 1 = (−3/2)(−2) + c, so c = −2: y = −3x/2 − 2.
  4. y = −x + 4 through (1, 5): new gradient 1. 5 = 1 + c, so c = 4: y = x + 4.

Worked example: perpendicular through a point

Find the equation of the line perpendicular to y = 3x − 1 that passes through (6, 4).

  1. The gradient of y = 3x − 1 is 3, so the new gradient is −1/3.
  2. Put (6, 4) in y = −x/3 + c: 4 = −2 + c, so c = 6.
  3. The equation is y = −x/3 + 6. Check: 3 × (−1/3) = −1, and at x = 6, y = −2 + 6 = 4.

Your go. The first step is given. Find the equation of the line parallel to 2x + y = 5 that passes through (1, 4).

  1. Rearrange 2x + y = 5 to y = −2x + 5, so the gradient is −2.
  2. A parallel line has the same gradient. Find c by putting (1, 4) in, then check.
Check the answer

4 = −2 × 1 + c gives c = 6, so y = −2x + 6. Check: at x = 1, y = 4.

Check before you go on

1. Which line is parallel to y = 3x − 2?

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. A line is parallel to y = 4x + 1 and passes through (2, 3). In y = 4x + c, what is c?

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

Level 2 · Building

4. A line is y = −0.25x + 3. What is the gradient of a line perpendicular to it?

5. A line is perpendicular to y = 4x + 3 and passes through (8, 1). In y = mx + c, what is c?

6. On a map grid a cycle lane runs along y = 0.5x + 1. A new lane crosses it at a right angle and passes through (2, 5). Where does the new lane cross the y-axis? Give the value of y.

Level 3 · Stretch

7. The lines 2x + 3y = 6 and 3x − ky = 8 are perpendicular. What is k?

8. A(1, 3), B(5, 5) and C(4, t) are points. The line BC is perpendicular to the line AB. Find t.

Something to think about

The line y = 3 is horizontal and x = 5 is vertical. Are they perpendicular? Does the rule m1 × m2 = −1 tell you so? What does that say about the rule?

One way to think about it

Yes, a horizontal line and a vertical line cross at a right angle. The rule cannot say so: the horizontal line has gradient 0, and 0 times anything is 0, not −1, and the vertical line has no gradient at all. The rule covers lines that have gradients that are not 0; these two cases need to be recognized on their own.

Exit ticket

9. Which pair of lines is perpendicular?