Verify geometric properties with algebra and coordinates

Ontario Grade 10 Mathematics

Analytic Geometry

Use points, slopes, lengths, and midpoints to check what a shape is

A drawing can suggest that two sides are parallel or that a shape is a rectangle. But a sketch may not be to scale. Coordinates let us check such claims with calculations. You will use familiar ideas about ordered pairs and subtraction, then apply three tools: slope, distance, and midpoint. Each tool answers a different question about a shape.

What you will learn

Find the slope, length, and midpoint of a line segment from its endpoint coordinates.
Use coordinate calculations to check whether sides are parallel or perpendicular.
Use evidence from coordinates to decide whether a quadrilateral has a stated property.

1. Grade 9 bridge: coordinates and change

A coordinate is an ordered pair such as

(3, 2)

. The first number tells how far to move horizontally from the origin. The second tells how far to move vertically. A point named

A(3, 2)

is located at those coordinates.

For two points, subtract their coordinates to find the horizontal and vertical changes. For example, moving from

(1, 2)

to

(5, 5)

gives a horizontal change of

4

and a vertical change of

3

. These changes help us describe the segment between the points.

The order matters when finding change: subtract the starting coordinate from the ending coordinate. If you reverse the order for both coordinates, the slope stays the same. Keep the order consistent to avoid sign errors.

An ordered pair gives a point's horizontal and vertical position.
Coordinate differences describe the movement from one point to another.

2. Three tools for checking a shape

The slope of a segment measures its steepness. Find it by dividing the vertical change by the horizontal change. Equal slopes show that two non-vertical lines are parallel. A horizontal line has slope

0

. A vertical line has an undefined slope because its horizontal change is

0

.

Two non-vertical lines are perpendicular when their slopes are negative reciprocals. This means one slope is the negative reciprocal of the other. For example, slopes

2

and

-\frac{1}{2}

describe perpendicular lines. A horizontal line and a vertical line are also perpendicular. Do not try to divide by zero to find the slope of a vertical line.

The distance formula finds the length of a segment from its endpoints. It comes from the right-triangle relationship you may know: the horizontal and vertical changes are the two shorter sides, and the segment is the hypotenuse. Equal lengths can help check whether sides or diagonals of a shape are congruent. Congruent segments have the same length.

The midpoint is the point exactly halfway between two endpoints. Find it by averaging the two horizontal coordinates and averaging the two vertical coordinates. If two diagonals have the same midpoint, they bisect each other: each diagonal cuts the other into two equal parts.

m=\frac{y_2-y_1}{x_2-x_1},\quad d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\quad M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Use slope to check parallel or perpendicular sides.
Use distance to compare segment lengths.
Use midpoint to check whether diagonals bisect each other.

3. Choose evidence that matches the claim

To verify a geometric property means to use calculations to check that the property is true for the given points. Start by identifying exactly what you need to check. A claim that opposite sides are parallel calls for slopes. A claim that all sides have equal length calls for distances. A claim that diagonals bisect each other calls for their midpoints.

A quadrilateral is a four-sided shape. Its vertices are the corner points, usually named in order around the shape. Its opposite sides do not share a vertex. Its diagonals join opposite vertices. For example, in quadrilateral

ABCD

, the sides are

AB

,

BC

,

CD

, and

DA

, and the diagonals are

AC

and

BD

.

For a parallelogram, opposite sides are parallel. So matching slopes for both pairs of opposite sides can verify this property. Another useful check is that the diagonals have the same midpoint. These checks use coordinate evidence rather than how the drawing looks. Choose a check that directly tests the stated property, and show enough calculations for someone else to follow.

Match the calculation to the property being checked.
A matching calculation supports the claim; the sketch alone does not.
Name the segments you compare so the evidence is clear.

4. Guided example and independent practice

Suppose the vertices of quadrilateral

ABCD

are

A(1,1)

,

B(5,3)

,

C(4,5)

, and

D(0,3)

, in order. We will check whether both pairs of opposite sides are parallel. Because parallel non-vertical lines have equal slopes, compare

AB

with

CD

, then

BC

with

DA

.

For

AB

, the vertical change is

3-1=2

, and the horizontal change is

5-1=4

. Its slope is

\frac{2}{4}=\frac{1}{2}

. For

CD

, use

C

as the starting point and

D

as the ending point. The changes are

3-5=-2

vertically and

0-4=-4

horizontally. Its slope is

\frac{-2}{-4}=\frac{1}{2}

. Thus,

AB

and

CD

are parallel.

For

BC

, the changes from

B

to

C

are

5-3=2

vertically and

4-5=-1

horizontally. Its slope is

-2

. For

DA

, the changes from

D

to

A

are

1-3=-2

vertically and

1-0=1

horizontally. Its slope is also

-2

. Thus,

BC

and

DA

are parallel. Both pairs of opposite sides are parallel, so the coordinates verify that

ABCD

is a parallelogram.

Try this on your own: For

P(0,0)

,

Q(6,0)

,

R(6,2)

, and

S(0,2)

, check whether opposite sides are parallel. Then use distances to check whether all four sides have equal length. Use the calculations, not the appearance of the points, to make each decision.

Keep the same endpoint order within each slope calculation.
A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
A separate claim, such as equal side lengths, needs its own evidence.

Choose a coordinate tool

Question	Tool	What to compare
Are two sides parallel?	Slope	Equal slopes, with vertical lines handled separately
Are two sides perpendicular?	Slope	Negative reciprocal slopes, or one horizontal and one vertical line
Are segments equal in length?	Distance	Their calculated lengths
Do diagonals bisect each other?	Midpoint	The midpoint of each diagonal

Worked example

Check a quadrilateral for parallel opposite sides

The vertices of

ABCD

are

A(1,1)

,

B(5,3)

,

C(4,5)

, and

D(0,3)

. Verify whether both pairs of opposite sides are parallel.

Identify the pairs
The opposite-side pairs are $AB$ with $CD$ , and $BC$ with $DA$ . Since the claim is about parallel sides, compare their slopes.
Compare the first pair
For each segment, subtract the starting coordinates from the ending coordinates. Both slopes are $\frac{1}{2}$ , so the non-vertical sides $AB$ and $CD$ are parallel.
$m_{AB}=\frac{3-1}{5-1}=\frac{1}{2},\quad m_{CD}=\frac{3-5}{0-4}=\frac{1}{2}$
Compare the second pair
The slopes of $BC$ and $DA$ are both $-2$ . These sides are also parallel.
$m_{BC}=\frac{5-3}{4-5}=-2,\quad m_{DA}=\frac{1-3}{1-0}=-2$
State the conclusion
Both pairs of opposite sides are parallel. That is the defining property being checked for a parallelogram, so the coordinates verify that $ABCD$ is a parallelogram.

Answer: Both pairs of opposite sides are parallel, so

ABCD

is a parallelogram.

Check: Each pair has equal slopes, and none of the four sides is vertical. The slope comparisons are therefore valid.

Common mistakes and how to avoid them

Subtracting the coordinates in different orders within one slope calculation.

Correction: Use the same endpoint order for the vertical and horizontal changes. Reversing both signs leaves the slope unchanged, but reversing only one changes it.

Calling a vertical line's slope zero.

Correction: A horizontal line has slope zero. A vertical line has undefined slope because its horizontal change is zero.

Using a drawing as the only reason that sides are parallel or equal.

Correction: Use slopes to check parallel sides and distances to compare lengths. A sketch may not be drawn to scale.

Using equal side lengths to claim that opposite sides are parallel.

Correction: Use a calculation that directly checks parallelism, such as comparing the slopes of opposite sides.

Lesson summary

Coordinates let you check shape properties instead of relying on a sketch.
Slope checks parallelism and perpendicularity; distance checks lengths; midpoint checks whether segments meet halfway.
Choose evidence that matches the exact property in the question.
For a parallelogram, both pairs of opposite sides are parallel.

Check your understanding

Question 1

A segment joins

(-2,1)

and

(4,1)

. What is its slope?

Undefined
0
1
6

Show answer and explanation

0

The vertical change is

1-1=0

, while the horizontal change is

4-(-2)=6

. The slope is

\frac{0}{6}=0

, so the segment is horizontal.

Question 2

Two non-vertical sides have slopes

\frac{3}{4}

and

-\frac{4}{3}

. What does this show?

The sides are parallel.
The sides are perpendicular.
The sides have equal lengths.
The sides have the same midpoint.

Show answer and explanation

The sides are perpendicular.

The slopes are negative reciprocals, so the lines are perpendicular. Slope alone does not show equal lengths or equal midpoints.

Question 3

Which tool should you use to check whether the diagonals of a quadrilateral bisect each other?

Compare their slopes only.
Compare the midpoints of the diagonals.
Compare the horizontal changes only.
Check whether one diagonal is longer.

Show answer and explanation

Compare the midpoints of the diagonals.

If both diagonals have the same midpoint, they cut each other into two equal parts. Comparing lengths or horizontal changes does not directly test this property.

Key terms

Coordinate: A number that gives a point's position on a grid.
Slope: A number that describes a line's steepness using vertical change divided by horizontal change.
Midpoint: The point exactly halfway between two endpoints.
Diagonal: A segment joining two opposite vertices of a polygon.
Verify: Check a statement using suitable calculations or evidence.

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