Coordinate Geometry

Mathematics revision notes · Cambridge (CIE) · 0580

1. Coordinates on the grid

The grid used throughout this course has a horizontal x-axis and a vertical y-axis, meeting at the origin
The origin is the point (0, 0), and every other point is described by how far it lies from there in each direction
A pair of coordinates is written in brackets with the horizontal value first: (x, y)
Positive x moves right and negative x moves left; positive y moves up and negative y moves down
- (4, 3) sits four to the right and three up
- (−2, −5) sits two to the left and five down
Plot a point with a small cross or a solid dot, and write its letter beside it when the question names it, so it is clear which point is which

Check the scale on the axes before counting: one square is frequently worth more than one unit, and assuming otherwise puts every answer out by a factor
The steepness of the line joining two points is dealt with separately, under linear graphs

The midpoint is the point exactly half way along, so it is the same distance from each end
Finding it means averaging the two ends one direction at a time: average the x values, then average the y values
For ends (x₁, y₁) and (x₂, y₂) the midpoint is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
This formula is not printed anywhere in the paper, so it has to be known
Each coordinate is worked out independently, and each one is worth a mark, so a slip in one does not cost the other
A half in the answer is perfectly normal and should be left as 0.5 or ½ rather than rounded away

Worked example

Midpoint from two given points

A is the point (−7, 2) and B is the point (3, 9). Find the coordinates of the midpoint of AB.

Solution:

A question may give one end and the midpoint and ask for the other end
Whatever the step is from the given end to the midpoint, the same step continues to the far end
In numbers, double the midpoint value and subtract the known end, in each direction separately
Checking is quick: average your answer with the given end and confirm the midpoint comes back

Worked example

Finding the far end

M(7, 4) is the midpoint of AB, and A is the point (3, −1). Find the coordinates of B.

Solution:

In the x direction: 2 × 7 − 3 = 14 − 3 = 11
In the y direction: 2 × 4 − (−1) = 8 + 1 = 9
So B is (11, 9)
Check: the midpoint of (3, −1) and (11, 9) is ((3 + 11) ÷ 2, (−1 + 9) ÷ 2) = (7, 4)

Worked example

A point that divides a segment in a ratio

A is the point (−1, 9) and B is the point (9, 4). P lies on AB with AP : PB = 3 : 2. Find the coordinates of P.

Solution:

The ratio 3 : 2 cuts AB into 3 + 2 = 5 equal parts, and P is 3 of them along from A (the midpoint is the 1 : 1 case)
From A to B, x changes by 9 − (−1) = 10 and y changes by 4 − 9 = −5
One part is 10 ÷ 5 = 2 in x and −5 ÷ 5 = −1 in y
Three parts on from A: x = −1 + 3 × 2 = 5 and y = 9 + 3 × (−1) = 6
So P is (5, 6)
Check: from P to B is 9 − 5 = 4 in x and 4 − 6 = −2 in y, which is two parts, so AP : PB = 3 : 2

The distance between two points is the hypotenuse of a right-angled triangle drawn with one horizontal and one vertical side
The horizontal side is the difference in the x values and the vertical side is the difference in the y values
So the length is √((x₂ − x₁)² + (y₂ − y₁)²)

Squaring removes any minus sign, so it does not matter which point is taken first
Like the midpoint formula, this one is not printed in the paper
Writing the two differences down before squaring them earns a mark on its own, and the completed substitution earns another
Put brackets around negative coordinates as you subtract them, since a subtraction of a negative is where the sign errors happen

Worked example

A length that is a whole number

C is the point (2, −5) and D is the point (14, 4). Calculate the length of CD.

Solution:

Where the number under the root is not a perfect square, the exact answer is a surd
Simplify it by taking out the largest square factor, exactly as in the surds topic
Questions on this specification regularly ask for the length in surd form, or state that it equals k√n and ask for k
Without a calculator, leave a length that is not a whole number as an exact surd; on the calculator paper a decimal correct to 3 significant figures (or to the accuracy the question gives) also scores, unless the question asks for a surd or an exact answer

Worked example

A length in surd form

A is the point (−1, 4) and B is the point (5, 1). Find the exact length of AB, writing the surd in its simplest form.

Solution: