Coordinate geometry: put geometry on a grid and let algebra do the work

How to use this simulation

1. Start in “Distance, midpoint, slope”. Two points, A and B, sit on the grid (they start at (-4, -2) and (4, 4)). Drag either one: they snap to whole-number positions. The dashed legs and the right-angle mark show the right triangle that hides behind the distance formula.
2. Read the numbers. Δx and Δy are the legs, “distance squared” is Δx² + Δy², and the distance is its square root. At the start AB = 10.000. The green point M is the midpoint, here (0, 1).
3. Make A and B level (same y) and the slope reads 0. Make them stacked (same x) and the slope says “undefined”, because a vertical line has no slope. Everything in between gives rise ÷ run.
4. Switch to “Dividing in a ratio”. Two sliders set m and n (they start at 2 and 3). The segment is cut into 5 equal steps and P sits 2 steps from A, at (-0.80, 0.40). Compare AP ÷ PB with m ÷ n in the readings; they match.
5. Tick “External division” to push P outside the segment. Then set m = n and see why the external point disappears: you would be dividing by zero.
6. Finish in “Locus (equidistant)”. A green line is the set of all points the same distance from A and B. Drag the orange point P across it: PA and PB are equal only when P sits exactly on the line, and the verdict tells you which side is closer.

A robot on a grid

Imagine a robot on a floor tiled with squares. You can tell it “go four tiles right and three up” and it knows exactly where to stop. Now ask a harder question: how far is its end point from the start in a straight line, not along the tiles? Or: where is the spot exactly halfway? Or: where is the spot one third of the way?

Coordinate geometry answers all of these without a ruler. The big idea, due to René Descartes in the 1600s, is to label every point with two numbers. Once a point is a pair of numbers, a line is an equation, a circle is an equation, and a geometry puzzle turns into arithmetic that a calculator can finish.

Coordinates from zero: axes, origin and quadrants

Draw a horizontal number line, the x-axis, and a vertical one, the y-axis. They cross at the origin, the point (0, 0). Every point on the plane gets an ordered pair (x, y): x tells you how far to move along the x-axis (right is positive, left is negative) and y tells you how far to move up or down.

Order matters. The point (3, 2) is three steps right and two steps up. The point (2, 3) is a different place. The two axes cut the plane into four quadrants, numbered anticlockwise starting from the top right.

The distance formula: Pythagoras in disguise

To find the straight-line distance between A(x₁, y₁) and B(x₂, y₂), do not measure anything. Instead walk from A to B in two moves: first sideways by Δx = x₂ − x₁, then up or down by Δy = y₂ − y₁. Those two moves are the legs of a right triangle, and the straight line AB is its hypotenuse.

The simulation draws that triangle with dashed legs and a right-angle mark. Pythagoras says the hypotenuse squared equals the sum of the legs squared, so take the square root and you have the distance. Squaring also removes any minus sign, which is why it does not matter which point you call A.

The midpoint: just average the coordinates

The midpoint M of AB is the point exactly halfway. Halfway in x is the average of the two x-values, and halfway in y is the average of the two y-values. There is nothing more to it.

In the simulation the green point M moves as you drag A or B. At the start it sits at (0, 1): add the x-values, halve; add the y-values, halve.

Slope: how steep is the line?

The slope m of the line through A and B is the rise divided by the run: how many units it climbs for every unit it moves right. Going up to the right gives a positive slope, going down gives a negative slope, a flat line has slope 0, and a vertical line has no slope at all because the run is 0 and you cannot divide by 0.

Slopes also tell you how two lines relate. Parallel lines have the same slope. Perpendicular lines have slopes whose product is −1, for example 2 and −1/2.

The section formula: cutting a segment in a ratio

Sometimes the halfway point is not what you want. A surveyor needs a flag post one third of the way along a boundary. A game designer needs an enemy to appear two fifths of the way from the gate to the tower. That is dividing the segment AB in the ratio m : n.

Imagine AB cut into m + n equal steps. The point P is m steps from A and n steps from B. Because the steps are equal, P is a weighted average of A and B: B counts m times and A counts n times, then divide by m + n. The simulation draws the tick marks so you can count the steps.

Locus: all the points equidistant from two points

A locus is the path of every point that obeys one rule. Take the rule “the same distance from A and from B”. Try some points: the midpoint M qualifies. So does a point straight above M, and one straight below. Join them all and you get a straight line through M at right angles to AB: the perpendicular bisector.

In the simulation, drag the orange point P. The two numbers PA and PB are the distances. When they are equal P is exactly on the green line, and the verdict turns green. On one side P is closer to A, on the other it is closer to B. A small green point Q glides along the line to show that QA and QB stay equal wherever it goes.

To get the equation, write PA² = PB² and expand. The squares x² and y² cancel and you are left with a straight line: Δx · x + Δy · y = (|B|² − |A|²) ÷ 2, where |B|² means x₂² + y₂². The readings print it for your points.

Try it in the simulation

Five small experiments turn the formulas into things you can feel. Do them with the handles and sliders, not just by reading.

• Drag B until Δx = 3 and Δy = 4. The distance is exactly 5. Now try Δx = 5, Δy = 12 for 13.
• Put A and B on the same horizontal line. The distance is just |Δx| and the slope is 0. Do the same on a vertical line and the slope becomes undefined.
• In “Dividing in a ratio”, set m = n. P lands on the midpoint. Increase m and watch P slide towards B.
• Tick “External division”, set m = 3 and n = 2. P jumps beyond B. AP ÷ PB is still m ÷ n, but P is outside the segment.
• In “Locus”, drag P along a diagonal line and watch PA and PB. They are equal at exactly one point on each line that crosses the bisector.

Solved problems

Each answer below is computed from the same functions the simulation uses.

Common mistakes

These slips cost marks every year. Check for them before you hand in a paper.

• Forgetting to square a negative difference. (−4)² is 16, not −16, and not 4.
• Subtracting in different orders for x and y in the distance formula. Squaring hides the sign, but the pairs must still be consistent.
• Using the wrong weights in the section formula. B pairs with m and A with n, not the other way round.
• Dividing by m − n in an external division when m = n. There is no such point.
• Writing the slope as run ÷ rise, or saying a vertical line has slope 0. It is undefined; a horizontal line has slope 0.

Where coordinates show up in real life

Every map app is coordinate geometry. Your phone converts your position into a coordinate pair, measures distances between pairs and finds midpoints for “meet in the middle” suggestions. Video games store every character as coordinates, and collision checks are the distance formula run thousands of times a second.

Engineers lay out bridges and roads on coordinate grids, and architects place walls and doors the same way. The section formula puts a flag at a third of the way along a boundary, and the locus idea decides where a new school should stand if it must be as close to one village as to another.

Computer graphics, robotics and even weather forecasting rest on the same idea: describe the world with numbers, then calculate.

Exam corner

Class 9 to 11 questions on coordinates are mostly one of five types. Learn to recognise the type before you start writing.

• Distance: find AB, or show that a triangle is isosceles, equilateral or right-angled by comparing distances.
• Midpoint: find the midpoint, or recover a missing endpoint or parameter.
• Slope: test for parallel, perpendicular or collinear points.
• Section formula: divide in a ratio, find the ratio, or find where an axis cuts a line.
• Locus: find the equation of the points equidistant from two points.

Revision summary

If you remember nothing else from this page, remember these lines.

• Distance is Pythagoras on the two legs Δx and Δy.
• Midpoint is the average; section is a weighted average.
• Parallel: equal slopes. Perpendicular: product −1.

Frequently asked questions