What is the gradient of a line perpendicular to one with gradient minus 2?

It is one half. The perpendicular gradient is the negative reciprocal: flip minus 2 to minus one half, then change the sign to one half. Check it with the product test: minus 2 times one half equals minus 1, which is the test for perpendicular lines. Keeping the sign (minus one half) or only changing the sign (2) both fail that test, and reusing minus 2 gives a parallel line.

What is the difference between parallel and perpendicular gradients?

Parallel lines have equal gradients, so a line parallel to one with gradient minus 2 also has gradient minus 2. Perpendicular lines have gradients that are negative reciprocals of each other and multiply to minus 1, so a line perpendicular to one with gradient minus 2 has gradient one half.

Do you read the gradient straight off any equation of a line?

Only once the equation is in the form y equals m x plus c, with a single y on the left. For 2y equals minus x plus 6 you must divide every term by 2 first, giving y equals minus one half x plus 3, so the gradient is minus one half, not minus 1.

GCSE Maths Higher · AQA · Coordinate geometry

Perpendicular and parallel gradients: use the negative reciprocal

For two lines to be perpendicular, their gradients must be negative reciprocals of each other: you flip the fraction AND change the sign, so the two gradients multiply to $-1$ . A line with gradient $-2$ has a perpendicular gradient of $\tfrac{1}{2}$ , because $(-2)\times\tfrac{1}{2} = -1$ .

The instinct is to relate the gradients with a half-remembered rule: reuse the same gradient (that is parallel, not perpendicular), flip the fraction but keep the sign, or change the sign without flipping. Each fails the product test. The fix is a fixed two-step habit, flip then negate, with a product-equals-minus-one check at the end.

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How to spot it in your own work

You gave a perpendicular line the same gradient, which actually makes it parallel.
You took the reciprocal but kept the sign, writing $-\tfrac{1}{2}$ for a line perpendicular to gradient $-2$ .
You changed the sign of the gradient but did not flip it, writing $2$ instead of $\tfrac{1}{2}$ .
You confused the two relationships: parallel is equal gradient, perpendicular is the negative reciprocal.

An exam question that triggers it

Here is the structure of JUN24 Paper 3 Question 15b:

$A$ is the point $(-5, 9)$ and $C$ is the point $(3, -7)$ .

Find the equation of the line perpendicular to $AC$ that passes through $C$ .

The gradient of $AC$ is $\frac{-7 - 9}{3 - (-5)} = \frac{-16}{8} = -2$ . The misconception reuses $-2$ and gives the parallel $y = -2x - 1$ . The fix takes the negative reciprocal $\tfrac{1}{2}$ and uses $C$ to get $y = \tfrac{1}{2}x - 8.5$ .

Why students fall for this

Both rules live in the same corner of memory, and under time pressure the wrong one fires. Parallel and perpendicular sound like a matched pair, so a student who recalls "do something to the gradient" can reach for equality, a flip, or a sign change with equal confidence.

The negative reciprocal is two operations bundled into one, flip and negate, and it is easy to perform only one of them. Flipping without negating, or negating without flipping, each leaves a plausible-looking gradient that is simply wrong.

The gradient itself is a second trap. Students learn that the coefficient of $x$ is the gradient, then apply that before the equation is in $y = mx + c$ form, so an equation like $2y = -x + 6$ hands back the wrong starting gradient.

The fix: Identify the relationship, then flip and negate the gradient

Step 1: identify the relationship. Parallel or perpendicular? They are not the same operation on the gradient.

Step 2: parallel means equal gradient. A line parallel to one with gradient $m$ also has gradient $m$ .

Step 3: perpendicular means flip AND negate. The perpendicular gradient is the negative reciprocal $-\tfrac{1}{m}$ : turn the fraction upside down and change its sign. So $-2$ becomes $\tfrac{1}{2}$ .

Step 4: sense-check the product is minus one. Perpendicular gradients satisfy $m_1 \times m_2 = -1$ . If the product is not $-1$ , you have flipped or negated only one, or reused the gradient.

Worked example

NOV23·1·24 structure: Line A has gradient $-2$ . Find the gradient of a perpendicular line B.

Flip and negate. Flip $-2$ to $-\tfrac{1}{2}$ , then change the sign to $\tfrac{1}{2}$ .
$(-2)\times\tfrac{1}{2} = -1$
Traps: $-\tfrac{1}{2}$ keeps the sign, $2$ negates only, $-2$ is parallel.

NOV24·2·17 structure: Line B has equation $2y = -x + 6$ . Find its gradient.

Rearrange first. Divide every term by 2: $y = -\tfrac{1}{2}x + 3$ , so the gradient is $-\tfrac{1}{2}$ . Reading $-1$ before dividing is the trap.

JUN24·3·15b structure: $A(-5, 9)$ , $C(3, -7)$ . Find the line perpendicular to $AC$ through $C$ .

Gradient of AC. $\frac{-7 - 9}{3 - (-5)} = \frac{-16}{8} = -2$ .
Negative reciprocal. Flip and negate $-2$ to $\tfrac{1}{2}$ .
Intercept through C.
$-7 = \tfrac{1}{2}(3) + c \;\Rightarrow\; c = -7 - 1.5 = -8.5$
So $y = \tfrac{1}{2}x - 8.5$ . Trap: reuse $-2$ , giving the parallel $y = -2x - 1$ .

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Common questions

What is the gradient perpendicular to $-2$ ?: $\tfrac{1}{2}$ . Flip $-2$ to $-\tfrac{1}{2}$ , then change the sign to $\tfrac{1}{2}$ , because perpendicular gradients multiply to $-1$ and $(-2)\times\tfrac{1}{2} = -1$ .
How is parallel different from perpendicular?: Parallel lines have equal gradients, so a line parallel to gradient $-2$ also has gradient $-2$ . Perpendicular lines have gradients that are negative reciprocals and multiply to $-1$ .
Can I read the gradient straight off $2y = -x + 6$ ?: No. Divide through by 2 first to get $y = -\tfrac{1}{2}x + 3$ . The coefficient of $x$ is the gradient only once the equation is in $y = mx + c$ form, so the gradient is $-\tfrac{1}{2}$ , not $-1$ .

Related misconceptions

Dividing a line in a ratioFinding the point that divides a line in a given ratio by taking the midpoint or stepping the wrong way along the segment, instead of moving the correct fraction of the way from the start point.
Reading a reordered linear equationMisreading the gradient and intercept of a line written out of the usual order, such as y = 5 - 3x, by taking the constant as the gradient or dropping the sign on the coefficient of x.

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