Coordinate formulae become easier to remember when their geometry is visible. Distance comes from Pythagoras, midpoint from averaging coordinates, and section formulae from proportional movement along a segment.
1. Distance formula from Pythagoras
For A(x₁,y₁) and B(x₂,y₂), horizontal change is x₂−x₁ and vertical change is y₂−y₁. Therefore
AB=√[(x₂−x₁)²+(y₂−y₁)²].
The order of subtraction does not matter after squaring, provided each coordinate difference uses the same two points consistently.
2. Worked distance example
For A(−2,3) and B(4,11), the changes are 6 and 8. Hence AB=√(36+64)=10.
3. Midpoint as an average
The midpoint of A(x₁,y₁) and B(x₂,y₂) is
((x₁+x₂)/2,(y₁+y₂)/2).
Each coordinate lies halfway between the corresponding endpoint coordinates.
4. Worked midpoint example
The midpoint of (−3,5) and (7,−1) is ((−3+7)/2,(5−1)/2)=(2,2).
5. Internal division of a segment
Suppose P divides AB internally in the ratio AP:PB=m:n. Then P lies m/(m+n) of the way from A to B.
Starting from A,
P=A+[m/(m+n)](B−A).
Coordinate-wise, this becomes P=((nx₁+mx₂)/(m+n),(ny₁+my₂)/(m+n)).
6. Worked section example
A=(1,2), B=(11,7), and AP:PB=2:3. Then P is 2/5 of the way from A to B.
B−A=(10,5), so P=(1,2)+(2/5)(10,5)=(5,4).
Check: AP has displacement (4,2), while PB has displacement (6,3). These are in ratio 2:3.
7. Midpoint is a special section case
For AP:PB=1:1, the section formula gives equal weighting and becomes the midpoint formula.
8. Use distance to prove geometric facts
To show triangle ABC is isosceles, one coordinate route is to calculate two side lengths and show equality.
Example: A(0,4), B(−3,0), C(3,0). AB=5 and AC=5, so AB=AC and the triangle is isosceles.
9. Use midpoint to prove bisection
If two diagonals of a quadrilateral have the same midpoint, then they bisect each other. This can help establish a parallelogram.
Coordinate proof should still state the geometric conclusion explicitly rather than stopping after two midpoint calculations.
10. Common mistakes
- Adding coordinate differences instead of using Pythagoras.
- Swapping the ratio weights in a section formula.
- Using midpoint when the ratio is not 1:1.
- Calculating equal values without stating the geometric conclusion they prove.
11. Practice
- Find the distance between (1,2) and (7,10).
- Find the midpoint of (−4,6) and (8,−2).
- A=(0,0), B=(15,10). Find P if AP:PB=2:3.
- Show that A(0,3), B(−4,0), C(4,0) form an isosceles triangle.
- Find the midpoint of the segment joining (2,−5) to (10,7).
12. Answers
- 10.
- (2,2).
- (6,4).
- AB=5 and AC=5, so AB=AC.
- (6,1).
Continue with the Coordinate Geometry guide or return to the Additional Mathematics Hub.