A line’s step
On a grid of dots, a straight line from one dot to another moves by a step: so many dots across and so many up or down. The line here goes 3 across and 2 up. Every piece of the line has that same direction, so any part of it that runs 3 across also rises 2.
Parallel: repeat the step
To draw a line parallel to it, start at any other dot and take exactly the same step: 3 across and 2 up. Both lines climb 2 for every 3 across, so they point in the same direction and the gap between them never changes. However far both are made longer, they never meet.
Count the step; do not judge it by eye. A line that goes 3 across and only 1 up looks close, but it is less steep, so the gap between the two lines changes and somewhere they meet. A step of 6 across and 4 up is the same direction as 3 across and 2 up, because it is the same step taken twice.
Both lines go 3 across and 2 up, from different starting dots, so they are parallel.
Perpendicular: swap the counts and flip one
To draw a line at a right angle to the first one, swap the two counts and flip the direction of one of them. The step 3 across and 2 up becomes 2 across and 3 down.
Here is why that makes a right angle. Draw the first step as a right-angled triangle: 3 along the bottom, 2 up the side, and the line itself as the sloping side. Now turn the whole triangle a quarter turn clockwise. The 3 across now points down, and the 2 up now points across, so the new triangle goes 2 across and 3 down. Its sloping side has turned through 90° with it, so the new line is perpendicular to the old one.
Flipping the other count instead, 2 back and 3 up, gives the same line traveled the other way. Both are perpendicular to the first.
The line goes 3 across and 2 up, then turns and goes 2 across and 3 down. The two parts meet at a right angle.
The usual mistakes
Swapping without flipping. 2 across and 3 up is steeper than 3 across and 2 up, but it still climbs, so it turns away from the first line by less than a right angle.
Flipping without swapping. 3 across and 2 down is the mirror image of the first step, not the first step turned a quarter turn, and it meets the first line at a slant, not at 90°.
Carrying straight on. Another 3 across and 2 up from the end of the line only makes the same line longer.
The same rules as gradients
A step of 3 across and 2 up is a gradient of : the line rises 2 for every 3 across. Parallel lines take the same step, so parallel lines have the same gradient.
The perpendicular step, 2 across and 3 down, has gradient : the fraction turned upside down, with its sign changed. Multiply the two gradients: . Perpendicular lines that are not vertical or horizontal always have gradients that multiply to −1.
m₁ = 0.68, m₂ = −1.47: turning the slope triangle a quarter turn swaps its run and rise and negates one of them, so m₂ = −1/m₁ and m₁ × m₂ = −1
Turn the line to 45° and read both gradients
The gold triangle is the gold line’s step. The green triangle is the same triangle turned a quarter turn, so its across and up swap places and one of them changes sign. Turn the gold line: the green line stays at right angles to it, and the two gradients always multiply to −1.
Stepping a set distance at a right angle
To move a set distance away from a line, move along a perpendicular. The line steps 3 across and 4 up, so a perpendicular step is 4 across and 3 down. By Pythagoras that step is long.
A move of 1 at right angles to the line is of that step: across and down. Moving straight down by 1 does not do it, because straight down is not at right angles to a sloping line.
Worked example: Two Lines of Paving Laid 2 m from the Walls of a Slanting Corner, and the Lamp Post Where They Cross
Question A courtyard lies between two straight walls that meet at a corner O. On a plan marked in meters, O is at (0, 0), one wall runs along the x-axis, the other runs along the line y = 34x, and the courtyard is the space between them for x > 0. A gardener lays two straight lines of paving stones inside the courtyard, each parallel to one wall and 2 m from it. (a) Find the equation of the line of paving that is parallel to the slanting wall. (b) A lamp post stands where the two lines of paving cross. Find its position, and its distance from the corner O.
1.The paving beside the x-axis is parallel to it and 2 m inside the courtyard, so it is the line y = 2.
The paving beside the x-axis is the line y = 2. 2.A perpendicular to the slanting wall has gradient −43, because 34 × (−43) = −1. Going 3 m across and 4 m down is a step of √32 + 42 = 5 m, so a step of 2 m is 25 of it: 1.2 m across and 1.6 m down. From the point (4, 3) on the wall, this reaches (5.2, 1.4), which is inside the courtyard and 2 m from the wall.
A step of 2 m at right angles to the wall, from (4, 3), is 1.2 m across and 1.6 m down: it reaches (5.2, 1.4). 3.(a) The paving has the wall's gradient, 34, and passes through (5.2, 1.4): y − 1.4 = 34(x − 5.2), so y = 34x − 3.9 + 1.4 = 34x − 2.5. Multiplied by 4 and rearranged, this is 3x − 4y = 10.
(a) The line through (5.2, 1.4) with the wall's gradient 34 is y = 34x − 2.5, that is 3x − 4y = 10. 4.The lamp post is on both lines of paving, so substitute y = 2: 2 = 34x − 2.5, so 34x = 4.5 and x = 6. The lamp post is at (6, 2).
The two lines of paving cross where 2 = 34x − 2.5: at (6, 2). 5.(b) Its distance from O is √62 + 22 = √40 = 2√10 ≈ 6.32 m. Check: stepping 2 m from the lamp post back toward the slanting wall, 1.2 m to the left and 1.6 m up, reaches (4.8, 3.6), and 34 × 4.8 = 3.6, so that point is on the wall.
(b) The lamp post at (6, 2) is √40 = 2√10 ≈ 6.32 m from O.
Answer: (a) y = 34x − 2.5, that is 3x − 4y = 10; (b) at (6, 2), 2√10 ≈ 6.32 m from O
Common mistakes
- Moving the wall's line 2 m straight down, to y = 34x − 2. That shift is measured vertically, not at right angles to the wall, and the lines y = 34x and y = 34x − 2 are only 2 × 45 = 1.6 m apart.
- Stepping 2 m from the slanting wall on the wrong side, which gives 3x − 4y = −10. That line lies above the slanting wall, outside the courtyard, and it meets y = 2 at x = −23, which is not in the courtyard at all.