Coordinate Geometry Proofs — Print

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

What the practice questions cover

These questions use 15 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Mean — Two kinds of centre: the balance point, and the middle value.
Bisect — Cut exactly in half.
Work backwards — Start from the result and undo each step.
Slope — How much y changes for each 1 that x increases.
Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
Distance formula — d = √(x2 - x1)2 + (y2 - y1)2.
Midpoint — The point halfway between two others: average the x values and average the y values.
Parallel — Same slope, different intercept.
Perpendicular — Slopes multiply to -1 — negative reciprocals.
Simplest radical form — No perfect square factors left under the root.
Segment — Two endpoints and everything between them. Written AB.
Prove — The statement you must arrive at. Always the last line.
Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
Diagonal — A segment joining two non-adjacent vertices.
Adjacent — The side next to that angle which is not the hypotenuse.

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.

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.

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.

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.

Worked examples

Proving a parallelogram

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

Slope AB = 4-25-1 = 12 One pair of opposite sides.
Slope DC = 8-67-3 = 12 — equal, so AB ∥ DC Same slope means parallel.
Slope BC = 8-47-5 = 2 and slope AD = 6-23-1 = 2 The other pair.
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?

A rectangle needs a right angle at a corner. It is already a parallelogram.
At B: slopes 12 and 2. The two sides meeting at B.
12 × 2 = 1, not -1. Perpendicular would require -1.
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.

Place it conveniently: (0,0), (a,0), (a,b), (0,b). Letters keep it general; zeros keep it short.
One diagonal runs (0,0) to (a,b): length √a²+b² Distance formula.
The other runs (a,0) to (0,b): length √a²+b² Same value.
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.

Problem 1

Find the slope of the segment from (1,2) to (5,4).

Worked answer

1/2

4-25-1 = 24 = 12

Problem 2

Find the length of the segment from (0,0) to (6,8).

Worked answer

√36+64 = √100

= 10

Problem 3

Find the midpoint of (2,4) and (8,10).

Worked answer

(5, 7)

(2+82, 4+102) = (5,7)

Problem 4

A line has slope 34. What slope is perpendicular to it?

Worked answer

-4/3

-43

Problem 5

You want to prove two sides are congruent. Which formula do you use?

Worked answer

distance

Congruent means equal length.

Use the distance formula.

Problem 6

You want to prove two sides are parallel. Which formula?

Worked answer

slope

Equal slopes mean parallel.

Problem 7

You want to prove the diagonals bisect each other. Which formula?

Worked answer

midpoint

If both diagonals have the same midpoint, they bisect each other.

Problem 8

Are the segments (1,2)–(5,4) and (3,6)–(7,8) parallel? Answer yes or no.

Worked answer

yes

Both have slope 24 = 12.

Equal slopes, so parallel.

Problem 9

Are the segments (0,0)–(2,4) and (0,0)–(4,-2) perpendicular? Answer yes or no.

Worked answer

yes

Slopes 2 and -12

2 × -12 = -1, so perpendicular.

Problem 10

A quadrilateral has both pairs of opposite sides with equal slopes. What is it?

Worked answer

parallelogram

Both pairs of opposite sides are parallel.

That is a parallelogram.

Problem 11

Find the exact length from (-3,4) to (2,-1) in simplest radical form.

Worked answer

5sqrt(2)

√25+25 = √50

= 5√2

Problem 12

Two sides meet where one is vertical and one is horizontal. Are they perpendicular?

Yes, though the slope product rule does not apply
No
Only if the lengths match
Cannot be determined

Worked answer

Yes, though the slope product rule does not apply

They meet at a right angle.

But undefined times 0 is not -1, so use the picture instead.

Problem 13

A(1,2), B(5,4), C(7,8), D(3,6). Is ABCD a rectangle? Answer yes or no.

Worked answer

Slope AB = 12, slope BC = 2

12 × 2 = 1, not -1

Not a right angle, so not a rectangle.

Problem 14

A(0,0), B(4,0), C(4,3), D(0,3). Find the length of a diagonal.

Worked answer

(0,0) to (4,3): √16+9 = 5

Problem 15

Prove the diagonals of the rectangle (0,0), (a,0), (a,b), (0,b) are congruent. Give the common length.

Worked answer

sqrt(a^2 + b^2)

(0,0) to (a,b): √a²+b²

(a,0) to (0,b): √a²+b²

Equal for every a and b.

Problem 16

The midpoint of AB is (4,5) and A is (1,2). Find B.

Worked answer

(7, 8)

1+x2 = 4, so x = 7

2+y2 = 5, so y = 8

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