MathematicsExam code: 0580

Coordinate Geometry

Coordinate Geometry and 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

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