Mathematics
Perpendicular: Lines That Meet at 90°
Perpendicular means meeting at exactly 90°. The right-angle square on a diagram, the gradients-multiply-to-minus-one test, and what a perpendicular height is.
Perpendicular
Two lines are perpendicular when they meet or cross at a right angle, exactly 90°.
- Also called
- At right angles, Normal, in later work on curves and surfaces
- Where students meet it
- Grade 5 mathematics for the right-angle meaning, and Grade 9 or 10 for the gradient test in coordinate geometry.
The short answer
Perpendicular means at right angles: two lines are perpendicular when they meet or cross at exactly 90°, marked on a diagram by a small square in the corner. On a graph, two lines are perpendicular when their gradients multiply to give −1.
An example
y = 2x + 1 and y = −½x + 4 → 2 × −½ = −1
The gradients multiply to −1, so the lines cross at right angles however far apart their intercepts are. To go the other way and construct a perpendicular gradient, turn the fraction upside down and change the sign: a gradient of ¾ gives −4⁄3, and a gradient of 5 gives −1⁄5.
The 90° meeting and the little square
Perpendicular is a relationship between two lines rather than a property of either one on its own. No single line is perpendicular; it is perpendicular to something. On a diagram the relationship is shown by a small square drawn in the corner where the lines meet, and that square is a statement of fact you are allowed to use.
The lines do not have to cross. Two segments meeting end to end in a T shape are perpendicular, and so are two segments that would meet at 90° if extended, even though as drawn they never touch.
In writing, the symbol is an upside-down T: AB ⊥ CD says that AB is perpendicular to CD, in the same way that a pair of upright bars says two lines are parallel.
The gradient test on a graph
Multiply the two gradients. If the answer is −1, the lines are perpendicular. This is the same thing as saying that one gradient is the negative reciprocal of the other, which is the phrase most textbooks use.
There is one pair the test cannot handle. A horizontal line has gradient 0 and a vertical line has no gradient at all, so no multiplication is possible — yet they clearly meet at 90°. Treat that pair as perpendicular by inspection rather than trying to force the arithmetic.
Perpendicular distance and perpendicular height
The word appears in formulae as often as it does in geometry questions. The area of a triangle is half the base times the perpendicular height, and the perpendicular height is measured at right angles to the base, not along a sloping side. Using the slanted length instead is the classic way to get a triangle's area slightly too big.
It carries the same meaning for the distance from a point to a line: the shortest route is the perpendicular one, which is why a question asking how far a boat is from the shore expects a right angle to be drawn.
Common questions
What is the gradient of a line perpendicular to y = 3x − 2?
−1⁄3. Write the gradient 3 as the fraction 3⁄1, turn it upside down to get 1⁄3, then change the sign. Multiplying back gives 3 × −1⁄3 = −1, which confirms it. Note that the −2 has no bearing on the answer, since it only moves the line up or down.
Do perpendicular lines have to cross each other?
No. They only need to meet at 90°, or to lie at 90° to one another when extended. The two sides of a rectangle that meet at a corner are perpendicular; so are the wall and the floor of a room, even though a short piece of skirting board touches only at the join.
Is perpendicular the same as vertical?
No. Vertical describes a single direction — straight up and down relative to the ground. Perpendicular describes how two lines sit relative to each other, and a pair of perpendicular lines can be tilted at any angle as long as the angle between them stays at 90°.
What does perpendicular height mean in an obtuse triangle?
The same thing, but it may fall outside the shape. When the triangle leans, the height measured at right angles to the base has to be taken from a base line extended beyond the triangle. The area formula is unchanged; only the drawing looks unfamiliar.
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Knowing the word is not the same as using it
A tutor can watch a student use this in a question and see exactly where the understanding stops. The first class is free.