Parallel and perpendicular lines | Year 11

01 / LEARN

The idea, step by step

Parallel lines

Distinct non-vertical lines are parallel when their gradients agree. They rise or fall at the same rate and never meet. If their intercepts also agree, the equations describe the same line, not two distinct parallel lines. Rewriting an equation in slope-intercept form makes this distinction clear.

All vertical lines are parallel to one another, even though their gradients are undefined. Coordinate geometry needs this exception; a rule involving finite slopes cannot be applied to a line of the form x=a.

Perpendicular lines

For two non-vertical, non-horizontal lines, perpendicular gradients multiply to -1. The perpendicular gradient is therefore the negative reciprocal of the original gradient. This condition assumes equal geometric units on the two axes; it is a coordinate statement rather than a judgement of a resized graph.

A horizontal line and a vertical line are perpendicular. If the original gradient is zero, use a vertical equation for the perpendicular line rather than dividing by zero. Once a slope is found, use the required point to determine the line completely.

m_{1} m_{2} = - 1

A common trap

Changing only the sign of the original gradient to find a perpendicular gradient.

Instead: Use the negative reciprocal. A gradient of 3 gives -1/3, not -3. Check that the product is -1.

02 / WORKED EXAMPLES

See the thinking

Example 1

A parallel line

Find the line parallel to $2 x - y = 7$ through (3,4).

Rearrange the given line to y=2x-7, so its gradient is 2.
Use $y - 4 = 2 (x - 3)$ , giving y=2x-2.

y = 2 x - 2

Example 2

A perpendicular line

Find the line perpendicular to $y = 2 x + 1$ through (4,-1).

The perpendicular slope is -1/2.
Use $y + 1 = - (x - 4) / 2$ and simplify.

y = - x / 2 + 1

Check your understanding

What gradient is perpendicular to -3/4?

Show answer

$4 / 3$ .

$(- 3 / 4) (4 / 3) = - 1$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Are $y = 3 x + 2$ and $6 x - 2 y = 5$ parallel, perpendicular or neither?
2 marks
Find a line parallel to $y = - 4 x + 9$ through (1,2).
2 marks
Find a line perpendicular to $y = x / 3 - 2$ through (2,5).
2 marks
Find the line perpendicular to y=7 through (-2,3).
2 marks
Determine whether the lines joining (0,0) to (2,6), and (1,4) to (7,2), are perpendicular.
2 marks
Two paths have equations $y = 2 x + 5$ and $4 x - 2 y = - 10$ . Are they distinct parallel paths?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Are $y = 3 x + 2$ and $6 x - 2 y = 5$ parallel, perpendicular or neither?
2 marks
The second is y=3x-2.5: same gradient, different intercept.
Parallel and distinct.
Find a line parallel to $y = - 4 x + 9$ through (1,2).
2 marks
$y - 2 = - 4 (x - 1)$ .
$y = - 4 x + 6$
Find a line perpendicular to $y = x / 3 - 2$ through (2,5).
2 marks
Perpendicular gradient -3; $y - 5 = - 3 (x - 2)$ .
$y = - 3 x + 11$
Find the line perpendicular to y=7 through (-2,3).
2 marks
The perpendicular to a horizontal line is vertical through the supplied x-coordinate.
$x = - 2$
Determine whether the lines joining (0,0) to (2,6), and (1,4) to (7,2), are perpendicular.
2 marks
The gradients are 3 and -1/3, whose product is -1.
Yes.
Two paths have equations $y = 2 x + 5$ and $4 x - 2 y = - 10$ . Are they distinct parallel paths?
2 marks
Rearranging the second gives y=2x+5 exactly.
No; they are the same line.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Topic 1: Functions and graphs. Prerequisite algebra and coordinate geometry are embedded as supporting lessons.

Official curriculum source ↗

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