Coordinate Geometry Proofs

Put a shape on axes and every claim about it becomes a calculation you can check.

Four points are plotted and joined up. It looks like a rectangle. Is it?

Looking is not evidence. A drawing can be a degree off and you would never see it.

is this really a rectangle, or does it just look like one? (2, 2) (6, 7) (10, 4) (6, -1) eyes are not evidence — but the four corners are numbers

But the corners are numbers, and numbers can settle it.

Find the four slopes. If opposite sides have equal slopes, they are parallel. If adjacent slopes multiply to -1, the corners are right angles.

Find the four lengths with the distance formula and you can tell a rectangle from a square.

So coordinate geometry converts "does it look like one?" into arithmetic. Three tools do nearly all of it: distance for lengths, slope for parallel and perpendicular, midpoint for bisecting.

is this really a rectangle, or does it just look like one? (2, 2) (6, 7) (10, 4) (6, -1) eyes are not evidence — but the four corners are numbers

Three tools, three jobs

Distance answers questions about length: is this side congruent to that one, is it isosceles, is it a rhombus.

Slope answers questions about direction: are these parallel, is that a right angle, is it a trapezoid.

Midpoint answers questions about bisecting: do the diagonals cut each other in half.

Pick the tool from the claim

"Congruent" needs distance. "Parallel" or "perpendicular" needs slope. "Bisects" needs midpoint. Read the thing you are proving and the tool is chosen for you.

Naming the shape

A parallelogram: both pairs of opposite sides have equal slopes. Or: the diagonals share a midpoint.

A rectangle: a parallelogram whose adjacent slopes multiply to -1. Or: whose diagonals are congruent.

A rhombus: a parallelogram with all four sides the same length. A square needs both tests.

Writing it up

State what you will show, do the arithmetic, then say what it proves. Three parts, in that order.

"AB and CD both have slope 34, so AB ∥ CD." The conclusion sentence is not optional — it is the proof.

Show all four sides or slopes when the claim is about all four. Two is not a proof about a quadrilateral.

Say what it means

A page of correct calculations with no concluding sentence scores badly. After each computation write the geometric fact it establishes.

Choosing convenient coordinates

When a proof is about any rectangle rather than a specific one, you may place it wherever you like.

Put one corner at the origin and one side along the x-axis: (0,0), (a,0), (a,b), (0,b).

The letters keep it general, and the zeros keep the algebra short. This is called a coordinate proof.

Placing a rectangle for a coordinate proof: anywhere, every corner carries p and q; with a corner at the origin the corners are (0, 0), (a, 0), (a, b) and (0, b) a proof about any rectangle may put it wherever the algebra is shortest placed anywhereevery corner drags p and q along corner at the originthe zeros keep the algebra short (p, q)(p + a, q) (p, q + b)(p + a, q + b) (0, 0)(a, 0) (a, b)(0, b) ab diagonal√(a² + b²) the letters keep it general — this is a coordinate proof

Perpendicular, carefully

Two lines are perpendicular when m1 × m2 = -1 — the slopes are negative reciprocals.

The exception is a horizontal and a vertical line: slopes 0 and undefined, which are perpendicular but whose product is not -1.

So check for vertical sides before reaching for the product rule.

Perpendicular lines with slopes 2/3 and −3/2 multiply to −1; a horizontal and a vertical line are the exception perpendicular slopes are negative reciprocals: they multiply to −1 slopes multiply to −1 the exception xy 32 2−3 m₁ =23 m₂ =−32 23×−32= −1 → perpendicular slope 0 slopeundefined perpendicular, but theproduct is not −1 so check for a vertical side before you reach for the product rule

Worked examples

Proving a parallelogram

Show A(1,2), B(5,4), C(7,8), D(3,6) is a parallelogram.

  1. Slope AB = 4-25-1 = 12 One pair of opposite sides.
  2. Slope DC = 8-67-3 = 12 — equal, so AB ∥ DC Same slope means parallel.
  3. Slope BC = 8-47-5 = 2 and slope AD = 6-23-1 = 2 The other pair.
  4. Both pairs of opposite sides are parallel, so ABCD is a parallelogram. The concluding sentence is the proof.

Parallelogram — both pairs of opposite sides parallel

Rectangle or not?

For that same quadrilateral, is it a rectangle?

  1. A rectangle needs a right angle at a corner. It is already a parallelogram.
  2. At B: slopes 12 and 2. The two sides meeting at B.
  3. 12 × 2 = 1, not -1. Perpendicular would require -1.
  4. So the angle at B is not right, and ABCD is not a rectangle. A parallelogram, and no more.

No — the slopes give 1, not -1

A general coordinate proof

Prove that the diagonals of any rectangle are congruent.

  1. Place it conveniently: (0,0), (a,0), (a,b), (0,b). Letters keep it general; zeros keep it short.
  2. One diagonal runs (0,0) to (a,b): length √a2+b2 Distance formula.
  3. The other runs (a,0) to (0,b): length √a2+b2 Same value.
  4. The two are equal for every a and b, so the diagonals of any rectangle are congruent. Because a and b were never pinned down, this covers all rectangles.

Both diagonals are √a2+b2

Words to know

Distance formula
d = √(x2-x1)2 + (y2-y1)2 — used to compare lengths.
Slope
m = y2-y1x2-x1 — used to test parallel and perpendicular.
Midpoint formula
(x1+x22, y1+y22) — used to test bisecting.
Perpendicular
Slopes multiply to -1; they are negative reciprocals.

Mistakes to watch for

Concluding from the picture. Four points can look like a rectangle and have a corner that is 88°. The numbers are the evidence, and a plot is only there to suggest what to test.

Checking two sides and claiming a quadrilateral. A claim about all four sides needs all four computed — one parallel pair only makes it a trapezoid.

Using the m1 m2 = -1 test on a vertical side. A vertical line has undefined slope, so the product rule breaks; a horizontal and a vertical side are perpendicular anyway. Check for vertical sides first.

Doing the arithmetic and stopping. "Both slopes are 12" is a fact about numbers; "so the sides are parallel" is the geometry. Without the sentence, nothing has been proved.

This is why the plane is called Cartesian — after Descartes, who published it in 1637.

Before it, geometry and algebra were separate subjects. Afterwards they were the same subject in two notations, and problems that had needed genius became routine.

Pierre de Fermat had the same idea slightly earlier but did not publish, which is a recurring theme in Fermat's life.

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