Parallel lines never meet
A straight line has no ends: the part we draw is only a piece of it. Two straight lines on a flat surface are parallel when they never meet, however far they are made longer in both directions. Parallel lines stay the same distance apart all the way along. The two rails of a straight railway track are parallel, and so are the lines on writing paper.
The top and bottom edges of a rectangle are parallel. Make both of them longer, to the left and to the right, and they never come any closer together. The same is true of the two side edges. On a diagram, parallel lines are marked with matching arrowheads, and AB ∥ CD is read "AB is parallel to CD".
A rectangle: the top and bottom edges are parallel, and so are the two side edges.
Direction, not length
Parallel is about direction. Two lines of the same length need not be parallel, and two parallel lines can be very different lengths, because what is drawn is only part of each line.
A trapezium shows both. Its top and bottom edges are parallel, even though the bottom is longer. Its two slanted sides lean toward each other: make them longer upward and they meet above the shape. So a trapezium has exactly one pair of parallel sides.
A trapezium: the top and bottom edges are parallel, and the two slanted sides would meet if they were made longer.
Perpendicular lines cross at a right angle
Two lines are perpendicular when they cross at a right angle, 90°. The corner is marked with a small square, and AB ⊥ CD is read "AB is perpendicular to CD".
When two lines cross, the angles next to each other on a straight line add up to 180°. So if one of the four angles is 90°, its neighbors are 180 − 90 = 90° as well, and the angle opposite it is also 90°. All four angles at a perpendicular crossing are right angles.
Perpendicular is about the angle between the two lines, not about which way they point on the page. Neither line has to be vertical or horizontal. Turn two perpendicular lines together to any slant, and they still cross at 90°.
Two lines meeting at a right angle, 90°, marked with a small square.
A square is both
In a square, the opposite sides are parallel: the top and bottom never meet, and neither do the two sides. The sides that meet at a corner are perpendicular, because every corner is a right angle. A rectangle is the same: two pairs of parallel sides, and perpendicular sides at every corner.
A square: opposite sides are parallel, and sides that meet at a corner are perpendicular.
Parallel lines on a graph
On a graph, the line y = mx + c has gradient m: it rises m for every 1 across. The lines and both have gradient , so both rise 1 for every 2 across. At every value of x the first line is above the second, so the gap between them never closes: they are parallel.
Lines with different gradients are not parallel. One rises faster than the other, so the gap between them changes, and somewhere the two lines cross.
and both rise 1 for every 2 across, and the first is always 3 above the second: they are parallel.
The gradient of a perpendicular line
Take the line . Its gradient is , so a step of 3 across and 1 up stays on it. A line at right angles to it comes from turning that step.
Turn the step a quarter turn, clockwise. The 3 across now points down, and the 1 up now points across, so the new step is 1 across and 3 down. The whole step has turned through 90°, so the line it makes is perpendicular to the first. Its gradient is −3 ÷ 1 = −3.
So the two numbers of the step swap places, and one of them changes sign. As gradients, becomes −3: turn the fraction upside down and change its sign. The two gradients multiply to , and the same is true of any two perpendicular lines that are not vertical or horizontal.
The gold line, , steps 3 across and 1 up. The white line, y = −3x + 11, steps 1 across and 3 down. They cross at (3, 2), at a right angle.
Worked example: A Drain Laid at Right Angles to a Road, and Where It Reaches a Stream
Question On a plan of a housing estate, 1 unit represents 10 m. A straight road follows the line y = 12x + 2, and a straight stream follows the line y = x − 7. A drain is to be laid from a gully at P(6, 5), at the side of the road, at right angles to the road, until it reaches the stream. (a) Find the equation of the line of the drain. (b) Find the point where the drain reaches the stream, and the length of the drain in meters.
1.The gully is on the road, since 12 × 6 + 2 = 5. The road's gradient is 12, so the drain's gradient m satisfies 12 × m = −1, and m = −2.
The road has gradient 12, so the drain has gradient −2, because 12 × (−2) = −1. 2.(a) The drain passes through (6, 5) with gradient −2: y − 5 = −2(x − 6), so y = −2x + 17.
(a) The drain through P(6, 5) with gradient −2 is y = −2x + 17. 3.The drain reaches the stream where both equations hold: −2x + 17 = x − 7, so 3x = 24 and x = 8. Then y = 8 − 7 = 1, and the drain ends at (8, 1).
It meets the stream where −2x + 17 = x − 7: x = 8. 4.On the plan the drain goes 2 units across and 4 units down, so its length is √22 + 42 = √20 = 2√5 units.
The drain ends at (8, 1), and on the plan it is √22 + 42 = 2√5 units long. 5.(b) The drain reaches the stream at (8, 1), and it is 2√5 × 10 = 20√5 ≈ 44.7 m long. Check: the drain falls 4 units for 2 across, a gradient of −2, and (8, 1) is on the stream because 8 − 7 = 1.
(b) At 10 m to a unit, the drain is 20√5 ≈ 44.7 m long.
Answer: (a) y = −2x + 17; (b) at (8, 1), and the drain is 20√5 ≈ 44.7 m long
Common mistakes
- Taking the drain's gradient as −12 or as 2, which makes only one of the two changes. The road's gradient must be turned upside down AND given the opposite sign, so that the two gradients multiply to −1: 12 × (−2) = −1.
- Giving the length as √20 ≈ 4.47 m. That is the length in plan units; each unit represents 10 m, so the drain is ten times as long.