Coordinate geometry proofs use the gradient, midpoint, and distance formulae to verify geometric properties of shapes. Instead of relying on a diagram, you place vertices at known coordinates and show algebraically that the shape has the properties it claims — for example, that opposite sides of a quadrilateral are parallel.
What three formulae do you need?
All coordinate geometry proofs use a combination of three formulae. Make sure you have them memorised.
| Formula | Used to show |
|---|---|
| Gradient = (y₂ − y₁) / (x₂ − x₁) | Lines are parallel (equal gradient) or perpendicular (gradients multiply to −1) |
| Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) | That a point bisects a line segment; diagonals bisect each other |
| Distance = √((x₂ − x₁)² + (y₂ − y₁)²) | Sides are equal in length; shape is a rhombus or square |
How do you prove two lines are parallel?
Two lines are parallel if and only if they have the same gradient.
Worked example: Show that the line through A(1, 2) and B(4, 8) is parallel to the line through C(0, −1) and D(3, 5).
Gradient of AB = (8 − 2) / (4 − 1) = 6 / 3 = 2
Gradient of CD = (5 − (−1)) / (3 − 0) = 6 / 3 = 2
Both gradients equal 2. Since the gradients are equal and the lines pass through different points, AB is parallel to CD. □
How do you prove two lines are perpendicular?
Two lines are perpendicular if the product of their gradients is −1.
Worked example: Show that PQ is perpendicular to QR, where P = (2, 5), Q = (6, 3), R = (7, 5).
Gradient of PQ = (3 − 5) / (6 − 2) = −2 / 4 = −½
Gradient of QR = (5 − 3) / (7 − 6) = 2 / 1 = 2
(−½) × 2 = −1 ✓
Since the product of the gradients is −1, PQ ⊥ QR. □
Proving a quadrilateral is a parallelogram
A parallelogram has both pairs of opposite sides parallel. Show that the gradients of each pair of opposite sides are equal.
Worked example: ABCD has vertices A(0, 0), B(4, 1), C(5, 4), D(1, 3). Prove ABCD is a parallelogram.
Gradient of AB = (1 − 0) / (4 − 0) = ¼
Gradient of DC = (4 − 3) / (5 − 1) = 1 / 4 = ¼ ✓ (AB ∥ DC)
Gradient of AD = (3 − 0) / (1 − 0) = 3
Gradient of BC = (4 − 1) / (5 − 4) = 3 / 1 = 3 ✓ (AD ∥ BC)
Both pairs of opposite sides are parallel, so ABCD is a parallelogram. □
Proving a quadrilateral is a rectangle
A rectangle is a parallelogram with right angles. After proving it is a parallelogram, show that adjacent sides are perpendicular (gradients multiply to −1).
Continued example: Is ABCD above a rectangle?
Gradient of AB = ¼, Gradient of AD = 3.
¼ × 3 = ¾ ≠ −1
So AB is not perpendicular to AD, and ABCD is not a rectangle (it is a parallelogram but not a rectangle).
For a rectangle, you would need to find coordinates where adjacent gradients multiply to −1.
Proving diagonals bisect each other
This proves a shape is a parallelogram using a different approach: if the diagonals share the same midpoint, each diagonal bisects the other, confirming a parallelogram.
Example: Show that the diagonals of ABCD (A(0,0), B(4,1), C(5,4), D(1,3)) bisect each other.
Midpoint of AC = ((0+5)/2, (0+4)/2) = (2.5, 2)
Midpoint of BD = ((4+1)/2, (1+3)/2) = (2.5, 2)
The diagonals share midpoint (2.5, 2), so they bisect each other — confirming ABCD is a parallelogram by a second method. □
Common mistakes
| Mistake | How to avoid |
|---|---|
| Subtracting coordinates in the wrong order | Always use (y₂ − y₁)/(x₂ − x₁) with the same subscript on top and bottom |
| Forgetting to check both pairs for a parallelogram | You must show both AB ∥ DC and AD ∥ BC |
| Confusing parallel (equal gradients) with perpendicular (product = −1) | Write the condition explicitly: "equal gradients → parallel" |
| Concluding "rectangle" from parallel sides alone | A rectangle needs both parallel sides AND right angles (perpendicular adjacent sides) |
Frequently asked questions
How do I prove a shape is a square?
A square is a rhombus with right angles, or equivalently a rectangle with equal sides. Show: (1) all four sides are equal length using the distance formula, and (2) adjacent sides are perpendicular using gradients multiplying to −1. Two conditions together prove a square.
What if I need to prove a shape is an isosceles triangle?
Calculate the lengths of all three sides using the distance formula. An isosceles triangle has exactly two sides of equal length. State which two sides are equal and their common length.
Do I need to use all three formulae in every proof?
No — use whichever formulae are relevant to the property you are proving. Proving parallel sides only needs gradients; proving diagonals bisect each other only needs midpoints; proving equal sides only needs distances. Read the question carefully to choose the right tool.
What makes a coordinate proof different from a standard geometric proof?
A standard geometric proof uses angle facts, congruence criteria, and known theorems (e.g. alternate angles on parallel lines). A coordinate proof assigns numerical coordinates to points and verifies properties by calculation. Both are valid; coordinate proofs are often more straightforward when exact coordinates are given.
Work through coordinate geometry proofs with Socratic hints from Professor Pi at aitutors.me.