Plan a multi-step coordinate proof of a geometric property

Ontario Grade 10 Mathematics

Analytic Geometry

Choose coordinates and calculations that make a geometric property clear

A coordinate proof uses points on a grid and calculations to support a statement about a shape. The main challenge is planning which calculations answer the question. First, recall two Grade 9 ideas: a point is described by an ordered pair, and the length of a segment depends on the horizontal and vertical changes between its endpoints. In this lesson, you will use those ideas to plan a proof about a rectangle.

What you will learn

Choose coordinates that represent a figure clearly.
Plan a sequence of calculations to prove a geometric property.
Explain how coordinate calculations support a conclusion.

1. Start with the property you need to show

A coordinate proof begins with a claim about a figure. For example, the claim might say that two segments have equal lengths or that two lines are parallel. A geometric property is a fact about a shape or its parts.

Before calculating, identify the exact property in the question. Then decide what evidence would support it. To show that two segments have equal lengths, calculate and compare their lengths. To show that two non-vertical lines are parallel, calculate and compare their slopes. Slope describes how much a line rises or falls as it moves horizontally.

A vertical line has no horizontal change, so its slope is undefined. Do not use the usual slope calculation for it. To show that two vertical segments lie on parallel lines, explain that both supporting lines are vertical and that the lines are distinct. A supporting line is the complete line that contains a segment. To establish that the lines are distinct, show that they have different horizontal coordinates. If one line is vertical and the other is not, they are not parallel.

This is the planning step: connect the property to a suitable calculation or comparison. Do not calculate every possible feature of the figure. If the claim is that diagonals have equal lengths, side slopes do not answer it; the diagonal lengths do.

Name the property before choosing a calculation.
Compare lengths to prove equal lengths.
Compare slopes only for non-vertical lines.
For two vertical segments, establish that their supporting lines are distinct as well as vertical.
Explain why the chosen evidence supports the claim.

2. Choose coordinates that simplify the work

Coordinates are written as an ordered pair, with the horizontal position first and the vertical position second. For example, the point

(3,2)

is 3 units right and 2 units up from the origin,

(0,0)

.

A useful coordinate setup places part of a figure along an axis or at the origin. This can make some coordinates zero and reduce arithmetic. The setup must still match the figure described in the question. For a rectangle, one corner can be placed at the origin, one side along the horizontal axis, and the adjacent side along the vertical axis. The other two sides then line up with those axes.

Letters can represent side lengths that may vary. For example, use

w

for a horizontal side length and

h

for a vertical side length. Side lengths are positive, so state that

w>0

and

h>0

.

The distance formula finds the length between two points. It uses the horizontal and vertical changes between the endpoints. The formula works for horizontal, vertical, and slanted segments. A proof plan should name the property, the points involved, the calculation needed, and what result would support the claim.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Use a coordinate setup that fits the figure and keeps calculations simple.
Use variables when the claim should apply to a range of side lengths.
The distance formula applies to vertical segments as well as other segments.

3. Link each step to the next

A multi-step proof has calculations that work together. Begin by stating how the figure is placed on the grid. Next, give the coordinates of the needed points. Then calculate the measurements that test the claim. Finish by stating what the comparison proves.

A planning table can help keep the purpose of each step clear. The table is not a proof by itself. The written explanation must still connect the calculation to the geometric claim.

Use variable names consistently. If

w

and

h

represent side lengths, keep those meanings throughout the proof. When two calculated lengths have the same expression, explain that this equality is the evidence for the stated property.

Give every calculation a clear role in the argument.
Keep point labels and variables consistent.
End by stating exactly what the calculations establish.

4. From a plan to a proof

A useful plan has four parts: name the claim, set up the coordinates, choose calculations that test the claim, and explain the conclusion. The guided example follows this sequence.

The example proves that the diagonals of a rectangle have equal lengths. A diagonal is a segment joining opposite vertices. Since the claim concerns equal lengths, we will calculate the length of each diagonal rather than compare slopes.

Choose evidence that directly answers the claim.
Use general side lengths to represent any rectangle in the chosen setup.
State the geometric conclusion after comparing the results.

A proof-planning checklist

Plan step	Question to ask	Purpose
Name the claim	What property must be shown?	Keeps the proof focused.
Choose coordinates	How can the figure fit the grid simply?	Reduces unnecessary arithmetic.
Choose evidence	What calculation or comparison tests the claim?	Connects the claim to the calculations.
Compare and conclude	What does the result establish?	Turns calculations into a geometric statement.

Worked example

Equal diagonals in a rectangle

Prove using coordinates that the diagonals of rectangle

ABCD

have equal lengths. Use

A(0,0)

,

B(w,0)

,

C(w,h)

, and

D(0,h)

, where

w>0

and

h>0

.

Choose the evidence
The claim is that diagonals $AC$ and $BD$ have equal lengths. Calculate the length of each diagonal from its endpoints and compare the results.
Calculate the first diagonal
For $AC$ , the horizontal change is $w-0$ and the vertical change is $h-0$ . Substitute these changes into the distance formula.
$AC=\sqrt{(w-0)^2+(h-0)^2}=\sqrt{w^2+h^2}$
Calculate the second diagonal
For $BD$ , the horizontal change is $0-w$ and the vertical change is $h-0$ . Squaring the horizontal change gives $w^2$ , so this diagonal has the same length expression.
$BD=\sqrt{(0-w)^2+(h-0)^2}=\sqrt{w^2+h^2}$
State the conclusion
Both diagonals have length $\sqrt{w^2+h^2}$ . Therefore, the diagonals have equal lengths. Since $w$ and $h$ represent any positive side lengths, the reasoning applies to the rectangle in this setup generally.
$AC=BD$

Answer: The diagonals of rectangle

ABCD

have equal lengths.

Check: Each diagonal uses a horizontal change with magnitude

w

and a vertical change with magnitude

h

. The distance formula squares both changes, so each length is

\sqrt{w^2+h^2}

.

Common mistakes and how to avoid them

Calculating slopes when the claim is about equal lengths.

Correction: Match the calculation to the property. Compare lengths to prove equal lengths.

Trying to calculate the slope of a vertical line.

Correction: A vertical line has undefined slope. To show two vertical supporting lines are parallel, establish that both are vertical and distinct. For example, show their horizontal coordinates differ.

Using one specific drawing when the claim is meant to cover all rectangles in the setup.

Correction: Use variables such as

w

and

h

for the side lengths, and state that they are positive.

Stopping after writing two calculations without stating the conclusion.

Correction: Compare the results in words and name the property they establish.

Changing a point’s coordinates without explaining the setup.

Correction: State how the figure is placed on the grid and label each vertex consistently.

Lesson summary

A coordinate proof connects a geometric claim to calculations using coordinates.
Plan by naming the property, choosing a useful setup, and selecting evidence that tests the property.
Use the distance formula to compare lengths. It works for vertical segments.
Use slope comparisons for non-vertical lines. To show two vertical supporting lines are parallel, establish that they are distinct as well as vertical.
Explain how the results establish the claim.

Check your understanding

Question 1

You need to prove that two segments have equal lengths. Which calculation is the best direct evidence?

Compare their lengths using endpoint coordinates.
Compare the vertical coordinates of one endpoint only.
Find the midpoint of one segment only.
Compare the names of the endpoints.

Show answer and explanation

Compare their lengths using endpoint coordinates.

The claim is about equal lengths, so calculate both lengths and compare the results.

Question 2

A rectangle has coordinates

P(0,0)

,

Q(a,0)

,

R(a,b)

, and

S(0,b)

. Which pair represents its diagonals?

$PQ$ and $QR$
$PQ$ and $RS$
$PR$ and $QS$
$PS$ and $QR$

Show answer and explanation

PR

and

QS

Diagonals join opposite vertices. In this labeling,

P

is opposite

R

, and

Q

is opposite

S

.

Question 3

In the worked example, why do the two diagonal lengths match?

Both use a horizontal change with magnitude $w$ and a vertical change with magnitude $h$ .
All segments in a rectangle have the same length.
The coordinates of every vertex are zero.
The diagonals lie along the coordinate axes.

Show answer and explanation

Both use a horizontal change with magnitude

w

and a vertical change with magnitude

h

.

The distance formula squares each change, so the sign of a change does not affect the result. Both lengths become

\sqrt{w^2+h^2}

.

Key terms

Coordinate: A number pair that gives a point’s horizontal and vertical position on a grid.
Diagonal: A segment joining two opposite vertices of a polygon.
Distance formula: A rule for finding the length between two points from their horizontal and vertical coordinate changes.
Slope: A measure of a line’s steepness, found by comparing vertical change with horizontal change; vertical lines have undefined slope.
Supporting line: The complete line that contains a segment.
Coordinate proof: A geometric argument that uses points on a grid and calculations to support a property.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G13. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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