Formula sheet
Midpoint
The midpoint formula is given on the exam formula sheet, so memorising it is not the point. You must know to use it whenever a question asks for the point halfway between two others (or the centre of a line segment or diameter), and be able to rearrange it when you are given the midpoint and one endpoint and asked to find the other.
What the symbols mean
- x₁, y₁ the coordinates of the first endpoint (x₁, y₁)
- x₂, y₂ the coordinates of the second endpoint (x₂, y₂)
- (x, y) the midpoint, the point exactly halfway between the two endpoints
Given in the exam, or memorise?
The midpoint formula is given on the exam formula sheet, so memorising it is not the point. You must know to use it whenever a question asks for the point halfway between two others (or the centre of a line segment or diameter), and be able to rearrange it when you are given the midpoint and one endpoint and asked to find the other.
Why it works
The midpoint is just the average of the two x-coordinates paired with the average of the two y-coordinates.
- Moving from one point to the other, the x-coordinate changes evenly, so halfway across is the mean of x₁ and x₂, that is (x₁ + x₂)/2.
- The same reasoning applies vertically, giving (y₁ + y₂)/2 for the y-coordinate.
- Pairing the two averages gives the coordinates of the midpoint.
Worked example 1
Find the midpoint of the line segment joining A(2, 3) and B(8, 7).
- Take (x₁, y₁) = (2, 3) and (x₂, y₂) = (8, 7).
- Midpoint = ((2 + 8)/2, (3 + 7)/2)
- = (10/2, 10/2)
- = (5, 5)
Worked example 2
M(4, −1) is the midpoint of AB. Given A is (1, 2), find the coordinates of B.
- Let B = (x, y). Using the x-coordinates: (1 + x)/2 = 4.
- So 1 + x = 8, giving x = 7.
- Using the y-coordinates: (2 + y)/2 = −1.
- So 2 + y = −2, giving y = −4.
Where students go wrong
- Subtracting the coordinates instead of adding, the midpoint uses (x₁ + x₂)/2, not (x₂ − x₁)/2 (that difference belongs to the distance and gradient formulas).
- Dividing only one coordinate by 2, or dividing the sum of all four numbers by 2, each axis is averaged separately.
- Sign slips with negatives, e.g. (3 + (−7))/2 = −4/2 = −2, not (3 + 7)/2 = 5.
- When finding a missing endpoint, forgetting to multiply the midpoint value by 2 before subtracting the known coordinate.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Can the midpoint have fractions or decimals?
Yes. The midpoint is an average, so it often lands on non-whole values.
For example, the midpoint of (1, 2) and (4, 5) is (2.5, 3.5). Leave it as a fraction or decimal exactly as the division gives, do not round it to whole numbers, because that would move the point.
How do I find one endpoint when I'm given the midpoint and the other endpoint?
Set the midpoint formula equal to the known midpoint, one axis at a time. If (x₁ + x)/2 equals the midpoint's x-value, multiply both sides by 2 and subtract the known coordinate to get x.
Repeat for y. Worked example two above shows the full steps.
Is the midpoint always in the middle of the line?
Yes, the formula gives the point exactly halfway along the straight segment joining the two points, splitting it into two equal lengths. It is also the centre of a circle when the two points are the ends of a diameter, which is a common way the formula is tested.