Polygons in the Coordinate Plane

A polygon is a closed two-dimensional figure made from straight line segments. When the vertices of a polygon are placed on a coordinate plane, we can use algebra to investigate its geometric properties.

Instead of deciding what a shape is simply by looking at it, coordinate geometry allows us to use mathematical evidence.

Three particularly useful tools are:

• slope — investigates parallel and perpendicular sides
• distance — investigates side lengths and diagonal lengths
• midpoint — investigates diagonals and bisection

Together, these tools allow us to classify polygons and verify their properties.

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Naming Polygons

Polygons are usually classified first by their number of sides.

A polygon with all sides and all angles equal is called a regular polygon.

A polygon that does not have all sides and angles equal is an irregular polygon.

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Vertices and Sides

A polygon is described using its vertices, which are the points where sides meet.

For quadrilateral ABCD:

• A, B, C, and D are vertices
• AB, BC, CD, and DA are sides
• AC and BD are diagonals

When coordinates are given for the vertices, these segments can be analyzed mathematically.

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Plotting Vertices in the Correct Order

Suppose a quadrilateral has vertices:

A(1, 1)

B(6, 1)

C(6, 5)

D(1, 5)

The vertices should normally be connected in order:

A → B → C → D → A

Connecting points in the wrong order can produce a completely different figure.

Always check the intended order before calculating side properties.

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The Three Main Coordinate Tools

The distance formula is:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The slope formula is:

m = (y₂ − y₁)/(x₂ − x₁)

The midpoint formula is:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Each answers a different geometric question.

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What Each Tool Tells Us

Use distance when you need to determine:

• whether sides are congruent
• whether diagonals are congruent
• whether a triangle is isosceles or equilateral
• whether a quadrilateral has four equal sides

Use slope when you need to determine:

• whether sides are parallel
• whether sides are perpendicular
• whether angles are right angles

Use midpoint when you need to determine:

• whether diagonals bisect each other
• whether a point bisects a segment
• whether diagonals share the same midpoint

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Classifying Triangles

Triangles can be classified by their:

• side lengths
• angle relationships

Using distance, a triangle may be classified as:

scalene — no equal sides

isosceles — at least two equal sides

equilateral — all three sides equal

Using slopes, we can also determine whether a triangle contains a right angle.

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Example: Classifying a Triangle by Side Length

Consider:

A(0, 0)

B(6, 0)

C(3, 4)

Find AB:

AB = 6

Find AC:

AC = √[(3 − 0)² + (4 − 0)²]

AC = √25

AC = 5

Find BC:

BC = √[(3 − 6)² + (4 − 0)²]

BC = √25

BC = 5

Therefore:

AC = BC

The triangle has two congruent sides.

Therefore:

triangle ABC is isosceles.

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Example: Identifying a Right Triangle

Consider:

A(1, 1)

B(6, 1)

C(6, 5)

AB is horizontal because both points have:

y = 1

BC is vertical because both points have:

x = 6

Horizontal and vertical lines are perpendicular.

Therefore:

AB ⟂ BC

and:

∠ABC = 90°

So ABC is a:

right triangle

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Classifying Quadrilaterals

Coordinate geometry is especially useful for classifying quadrilaterals.

Important quadrilaterals include:

• parallelogram
• rectangle
• rhombus
• square
• trapezoid
• kite

Each has particular geometric properties that can be tested using coordinates.

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Parallelograms

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

For quadrilateral ABCD, we can prove it is a parallelogram by showing:

AB ∥ CD

and:

BC ∥ AD

Using coordinates, compare the slopes of opposite sides.

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Example: Identifying a Parallelogram

Consider:

A(1, 1)

B(6, 1)

C(8, 4)

D(3, 4)

Slope AB:

m_AB = 0

Slope CD:

m_CD = 0

Therefore:

AB ∥ CD

Slope BC:

m_BC = (4 − 1)/(8 − 6)

m_BC = 3/2

Slope AD:

m_AD = (4 − 1)/(3 − 1)

m_AD = 3/2

Therefore:

BC ∥ AD

Both pairs of opposite sides are parallel.

Therefore:

ABCD is a parallelogram.

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Another Test for a Parallelogram

The diagonals of a parallelogram:

bisect each other

Therefore, another coordinate test is to find the midpoints of both diagonals.

If AC and BD have the same midpoint, then the diagonals bisect each other.

This is sufficient to prove that a quadrilateral is a parallelogram.

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Parallelogram Using Midpoints

Using:

A(1, 1)

B(6, 1)

C(8, 4)

D(3, 4)

Midpoint of AC:

M_AC = ((1 + 8)/2, (1 + 4)/2)

M_AC = (4.5, 2.5)

Midpoint of BD:

M_BD = ((6 + 3)/2, (1 + 4)/2)

M_BD = (4.5, 2.5)

The diagonals have the same midpoint.

Therefore, they bisect each other.

Thus:

ABCD is a parallelogram.

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Rectangles

A rectangle is a parallelogram with four right angles.

Useful properties include:

• opposite sides are parallel
• opposite sides are congruent
• adjacent sides are perpendicular
• diagonals bisect each other
• diagonals are congruent

To classify a coordinate figure as a rectangle, slope is especially useful.

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Example: Identifying a Rectangle

Consider:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

AB and CD are horizontal.

BC and AD are vertical.

Therefore:

AB ∥ CD

BC ∥ AD

The figure is a parallelogram.

Also:

AB ⟂ BC

so it contains a right angle.

A parallelogram with a right angle is a rectangle.

Therefore:

ABCD is a rectangle.

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Rotated Rectangles

Rectangles do not need to have horizontal and vertical sides.

Consider:

A(0, 0)

B(4, 2)

C(3, 4)

D(−1, 2)

Slope AB:

m_AB = 2/4 = 1/2

Slope BC:

m_BC = (4 − 2)/(3 − 4)

m_BC = 2/(−1)

m_BC = −2

Since:

(1/2)(−2) = −1

AB and BC are perpendicular.

The other opposite sides have matching slopes.

Therefore, the figure is a rectangle even though it is rotated.

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Rhombuses

A rhombus is a parallelogram with four congruent sides.

To prove that a coordinate figure is a rhombus, distance is especially useful.

Calculate:

AB

BC

CD

DA

If all four lengths are equal, and the vertices form a proper quadrilateral, the figure is a rhombus.

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Example: Identifying a Rhombus

Consider:

A(0, 3)

B(4, 0)

C(8, 3)

D(4, 6)

Rather than calculating square roots, compare squared distances.

AB²:

(4 − 0)² + (0 − 3)²

= 16 + 9

= 25

BC²:

(8 − 4)² + (3 − 0)²

= 16 + 9

= 25

CD²:

(4 − 8)² + (6 − 3)²

= 16 + 9

= 25

DA²:

(0 − 4)² + (3 − 6)²

= 16 + 9

= 25

Therefore:

AB = BC = CD = DA

The quadrilateral has four congruent sides.

Therefore:

ABCD is a rhombus.

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Squares

A square is both:

• a rectangle
• a rhombus

It has:

• four congruent sides
• four right angles
• opposite sides parallel
• congruent diagonals
• diagonals that bisect each other

Coordinate geometry can verify these properties using distance, slope, and midpoint.

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Example: Identifying a Square

Consider:

A(0, 0)

B(4, 0)

C(4, 4)

D(0, 4)

Side lengths:

AB = 4

BC = 4

CD = 4

DA = 4

Therefore, all four sides are congruent.

AB is horizontal.

BC is vertical.

Therefore:

AB ⟂ BC

The quadrilateral has four equal sides and a right angle.

Therefore:

ABCD is a square.

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A Rotated Square

Consider:

A(0, 3)

B(3, 0)

C(6, 3)

D(3, 6)

Squared side lengths are:

AB² = 18

BC² = 18

CD² = 18

DA² = 18

Therefore, all sides are congruent.

Now calculate two adjacent slopes:

m_AB = (0 − 3)/(3 − 0)

m_AB = −1

m_BC = (3 − 0)/(6 − 3)

m_BC = 1

Their product is:

(−1)(1) = −1

Therefore:

AB ⟂ BC

The figure has equal sides and a right angle.

Therefore:

ABCD is a square.

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Why Appearance Can Be Misleading

A polygon may look like a square or rectangle on a coordinate grid but fail to satisfy the required properties.

For example, a quadrilateral might appear to have right angles, but its adjacent slopes could be:

2/3 and −1

Since:

(2/3)(−1) ≠ −1

the sides are not perpendicular.

Therefore, the apparent angle is not exactly 90°.

Coordinate calculations provide more reliable evidence than appearance.

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Trapezoids

A trapezoid is commonly defined in school geometry as a quadrilateral with at least one pair of parallel sides, although some curricula use the definition exactly one pair.

Check the convention being used in your course.

Suppose:

A(0, 0)

B(6, 0)

C(4, 3)

D(1, 3)

AB is horizontal.

CD is horizontal.

Therefore:

AB ∥ CD

The other two sides are not parallel.

Under the exactly-one-pair convention:

ABCD is a trapezoid.

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Isosceles Trapezoids

An isosceles trapezoid has congruent nonparallel sides.

Using coordinates:

1. prove one pair of opposite sides is parallel
2. calculate the lengths of the nonparallel sides
3. show those lengths are equal

For example, if:

AB ∥ CD

and:

AD = BC

then the figure may be classified as an isosceles trapezoid.

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Kites

A kite has two pairs of adjacent congruent sides.

For quadrilateral ABCD, a kite might satisfy:

AB = AD

and:

BC = CD

Distance calculations can verify these relationships.

Unlike a rhombus, the four sides of a kite do not all need to be equal.

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Polygon Classification Can Overlap

Some polygons belong to more than one category.

For example, a square is also:

• a rectangle
• a rhombus
• a parallelogram
• a quadrilateral

This creates a hierarchy of polygon classifications.

A square satisfies all the properties required for each of these broader categories.

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Most Specific Classification

Suppose a quadrilateral has:

• four congruent sides
• four right angles

Calling it a parallelogram is technically correct.

Calling it a rectangle is also correct.

Calling it a rhombus is also correct.

But the most specific classification is:

square

When asked to classify a polygon, give the most specific category supported by the evidence.

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Using Diagonals

Diagonals provide additional ways to classify quadrilaterals.

Useful relationships include:

Parallelogram: diagonals bisect each other.

Rectangle: diagonals bisect each other and are congruent.

Rhombus: diagonals bisect each other and are perpendicular.

Square: diagonals are congruent, perpendicular, and bisect each other.

These properties can be tested using distance, midpoint, and slope.

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Testing Whether Diagonals Bisect Each Other

Suppose:

A(0, 0)

B(6, 0)

C(8, 4)

D(2, 4)

Diagonal AC has midpoint:

((0 + 8)/2, (0 + 4)/2)

= (4, 2)

Diagonal BD has midpoint:

((6 + 2)/2, (0 + 4)/2)

= (4, 2)

Because the midpoints are equal:

the diagonals bisect each other

This supports the conclusion that ABCD is a parallelogram.

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Testing Whether Diagonals Are Congruent

Suppose a quadrilateral has diagonals AC and BD.

Calculate:

AC

and:

BD

If:

AC = BD

then the diagonals are congruent.

However, congruent diagonals alone are usually not enough to classify a quadrilateral as a rectangle.

The property should be combined with other information, such as showing that the figure is already a parallelogram.

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Testing Whether Diagonals Are Perpendicular

Calculate the slopes of the diagonals.

If:

m_AC × m_BD = −1

then the diagonals are perpendicular, provided both slopes are defined and nonzero.

A horizontal diagonal and vertical diagonal are also perpendicular.

Perpendicular diagonals can provide evidence for a rhombus, kite, or square, depending on the other properties.

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A Complete Classification Example

Consider:

A(1, 1)

B(5, 3)

C(3, 7)

D(−1, 5)

First calculate squared side lengths.

AB²:

(5 − 1)² + (3 − 1)²

= 16 + 4

= 20

BC²:

(3 − 5)² + (7 − 3)²

= 4 + 16

= 20

CD²:

(−1 − 3)² + (5 − 7)²

= 16 + 4

= 20

DA²:

(1 − (−1))² + (1 − 5)²

= 4 + 16

= 20

All four sides are congruent.

Now calculate adjacent slopes:

m_AB = (3 − 1)/(5 − 1)

m_AB = 1/2

m_BC = (7 − 3)/(3 − 5)

m_BC = 4/(−2)

m_BC = −2

Since:

(1/2)(−2) = −1

AB ⟂ BC.

The quadrilateral has:

• four congruent sides
• a right angle

Therefore:

ABCD is a square.

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Classifying Without Doing Every Calculation

A complete classification does not always require every possible calculation.

For example, once you establish that a quadrilateral is a parallelogram, you might need only one additional property:

Parallelogram + one right angle → rectangle

Parallelogram + four equal sides → rhombus

Rhombus + one right angle → square

Rectangle + four equal sides → square

Efficient reasoning avoids unnecessary work.

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Using Squared Distances

When comparing lengths, square roots are often unnecessary.

Suppose:

AB² = 37

BC² = 37

CD² = 37

DA² = 37

Then:

AB = BC = CD = DA

There is no need to calculate √37 four times.

Squared distances are especially useful when classifying polygons.

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Finding a Missing Vertex

Coordinate relationships can sometimes be used to determine a missing vertex.

Suppose ABCD is a parallelogram with:

A(1, 2)

B(6, 2)

C(8, 6)

and D is unknown.

Because opposite sides of a parallelogram have the same displacement:

From B to C:

right 2, up 4

Apply the same displacement from A to D:

(1 + 2, 2 + 4)

Therefore:

D = (3, 6)

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Finding a Missing Vertex Using Diagonal Midpoints

The same problem can be solved using the fact that parallelogram diagonals bisect each other.

For:

A(1, 2)

B(6, 2)

C(8, 6)

find the midpoint of AC:

M_AC = ((1 + 8)/2, (2 + 6)/2)

M_AC = (4.5, 4)

The midpoint of BD must also be:

(4.5, 4)

If D = (x, y):

(6 + x)/2 = 4.5

6 + x = 9

x = 3

and:

(2 + y)/2 = 4

2 + y = 8

y = 6

Therefore:

D = (3, 6)

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Finding Perimeter

Once a polygon's vertices are known, its perimeter can be calculated by finding the length of every side and adding them.

For a quadrilateral:

P = AB + BC + CD + DA

For polygons with more sides, continue adding each side length.

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Perimeter Example

Suppose a rectangle has vertices:

A(1, 1)

B(7, 1)

C(7, 5)

D(1, 5)

Lengths:

AB = 6

BC = 4

CD = 6

DA = 4

Therefore:

P = 6 + 4 + 6 + 4

P = 20 units

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Area of Coordinate Polygons

Simple coordinate polygons can often have their area calculated using familiar formulas.

For a rectangle:

A = length × width

For a triangle:

A = 1/2 × base × height

For a parallelogram:

A = base × perpendicular height

For more complex polygons, the figure can sometimes be divided into simpler shapes.

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Area Example

A rectangle has vertices:

(2, 3)

(10, 3)

(10, 8)

(2, 8)

Length:

10 − 2 = 8

Width:

8 − 3 = 5

Therefore:

A = 8 × 5

A = 40 square units

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Irregular Polygons

Not every polygon fits a special category.

An irregular quadrilateral might have:

• no parallel sides
• no equal sides
• no right angles
• diagonals with no special relationship

Coordinate geometry still allows us to describe it accurately.

We can calculate:

• side lengths
• slopes
• perimeter
• area
• diagonal lengths
• midpoint relationships

A figure does not need a special name to be mathematically analyzed.

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Concave and Convex Polygons

A convex polygon has no inward indentation. Every interior angle is less than 180°.

A concave polygon contains at least one inward indentation and at least one interior angle greater than 180°.

Coordinate diagrams can contain either type.

When plotting a polygon, the order of the vertices is particularly important for recognizing whether it is convex or concave.

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Coordinate Geometry in Design

Coordinate polygons can model:

• floor plans
• property boundaries
• computer graphics
• mechanical components
• maps
• structural designs
• robotic work areas
• game environments

Coordinates allow designers to verify dimensions and relationships precisely.

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Example: Checking a Rectangular Floor Plan

Suppose the corners of a room are represented by:

A(2, 1)

B(10, 5)

C(8, 9)

D(0, 5)

A drawing may make the room look rectangular.

Calculate:

m_AB = (5 − 1)/(10 − 2)

m_AB = 4/8

m_AB = 1/2

m_BC = (9 − 5)/(8 − 10)

m_BC = 4/(−2)

m_BC = −2

Their product is:

(1/2)(−2) = −1

Therefore:

AB ⟂ BC

The opposite sides can then be checked for parallelism.

Coordinate geometry allows the rectangular design to be verified mathematically rather than visually.

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Example: Verifying a Square Design

Suppose a designer specifies:

A(1, 4)

B(4, 1)

C(7, 4)

D(4, 7)

Squared side lengths:

AB² = 18

BC² = 18

CD² = 18

DA² = 18

Therefore, all sides are equal.

Slope AB:

m_AB = −1

Slope BC:

m_BC = 1

Therefore:

AB ⟂ BC

The design has equal sides and right angles.

Therefore:

the polygon is a square.

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Coordinate Classification Strategy

When given a set of polygon vertices, use this process:

1. Plot or visualize the points in order.
2. Determine the number of sides.
3. Identify which properties might be useful.
4. Calculate side slopes.
5. Compare slopes for parallelism.
6. Check adjacent slopes for perpendicularity.
7. Calculate side lengths or squared lengths.
8. Compare lengths for congruence.
9. Find diagonal midpoints if useful.
10. Examine diagonal lengths or slopes if necessary.
11. Match the evidence to polygon definitions.
12. Give the most specific classification justified by the evidence.

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A Quadrilateral Classification Checklist

For a quadrilateral, ask:

Are opposite sides parallel?

If yes, investigate parallelogram-family shapes.

Are adjacent sides perpendicular?

If yes, investigate rectangles and squares.

Are all four sides equal?

If yes, investigate rhombuses and squares.

Do the diagonals bisect each other?

This supports a parallelogram classification.

Are the diagonals congruent?

Combined with parallelogram properties, this supports a rectangle.

Are the diagonals perpendicular?

Combined with appropriate parallelogram properties, this supports a rhombus.

The final classification should use all relevant evidence.

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Verifying Rather Than Assuming

Suppose a problem states only the coordinates and asks:

What type of quadrilateral is ABCD?

Do not begin by assuming that it is a rectangle or square.

Instead:

1. calculate properties
2. identify relationships
3. compare those relationships with definitions
4. classify the figure

The classification should be the result of the reasoning, not the starting assumption.

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A Reliable Strategy for Distance

For endpoints:

(x₁, y₁)

and:

(x₂, y₂)

calculate:

d² = (x₂ − x₁)² + (y₂ − y₁)²

If you are only comparing lengths, keep the squared distances.

For example:

AB² = 20

BC² = 20

immediately gives:

AB = BC

Take square roots only if the actual length is needed.

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A Reliable Strategy for Slope

For each relevant side:

m = (y₂ − y₁)/(x₂ − x₁)

Then compare.

Equal finite slopes:

parallel direction

Negative reciprocal slopes:

perpendicular

Horizontal and vertical lines:

perpendicular

Two distinct vertical lines:

parallel

Always watch for vertical-line special cases.

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A Reliable Strategy for Midpoints

For a segment with endpoints:

(x₁, y₁)

and:

(x₂, y₂)

use:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

For quadrilateral diagonals:

• find midpoint of AC
• find midpoint of BD
• compare them

If the midpoints are equal:

the diagonals bisect each other

This can provide a quick parallelogram test.

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Common Mistakes

Classifying by appearance

A shape that looks square may not have equal sides or right angles.

Connecting vertices in the wrong order

This can produce an entirely different polygon.

Using only one property

One pair of parallel sides does not prove a quadrilateral is a parallelogram.

Assuming equal sides imply a square

Four equal sides establish a rhombus. A square also requires right angles.

Assuming right angles imply a square

A rectangle does not need four equal sides.

Ignoring vertical slopes

Vertical lines have undefined slope.

Forgetting the negative reciprocal

The perpendicular slope to 2/3 is −3/2.

Calculating unnecessary square roots

Squared distances are enough when only comparing lengths.

Assuming congruent diagonals alone prove a rectangle

Additional information is needed.

Assuming perpendicular diagonals alone prove a rhombus

Other quadrilaterals, such as some kites, can also have perpendicular diagonals.

Ignoring polygon hierarchy

A square is also a rectangle, rhombus, parallelogram, and quadrilateral.

Giving a less specific classification

If the evidence proves a square, "parallelogram" is true but not the most informative answer.

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Key Terms

Polygon — closed two-dimensional figure made from straight line segments.

Vertex — point where two sides of a polygon meet.

Side — line segment forming part of a polygon's boundary.

Diagonal — segment joining two nonadjacent vertices.

Regular polygon — polygon with all sides and all angles congruent.

Irregular polygon — polygon that does not have all sides and angles congruent.

Triangle — three-sided polygon.

Quadrilateral — four-sided polygon.

Parallelogram — quadrilateral with both pairs of opposite sides parallel.

Rectangle — parallelogram with four right angles.

Rhombus — parallelogram with four congruent sides.

Square — quadrilateral with four congruent sides and four right angles.

Trapezoid — quadrilateral with at least one pair of parallel sides under the inclusive definition; some courses use exactly one pair.

Kite — quadrilateral with two pairs of adjacent congruent sides.

Congruent segments — segments having equal lengths.

Parallel lines — distinct coplanar lines that never intersect.

Perpendicular lines — lines meeting at a 90° angle.

Midpoint — point dividing a segment into two equal parts.

Convex polygon — polygon without an inward indentation.

Concave polygon — polygon containing an inward indentation.

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Key Takeaways

• Coordinate geometry allows polygons to be classified using mathematical evidence rather than appearance.
• Distance is useful for comparing side and diagonal lengths.
• Slope is useful for identifying parallel and perpendicular sides.
• Midpoints are useful for analyzing diagonals and bisection.
• Squared distances can often be compared without calculating square roots.
• A parallelogram has two pairs of opposite parallel sides.
• Diagonals that bisect each other provide another useful parallelogram test.
• A rectangle is a parallelogram with right angles.
• A rhombus is a parallelogram with four congruent sides.
• A square has four congruent sides and four right angles.
• A square is also a rectangle, rhombus, parallelogram, and quadrilateral.
• Diagonal properties can provide additional evidence when classifying quadrilaterals.
• Rotated polygons have the same geometric properties as axis-aligned polygons.
• Horizontal and vertical sides are useful special cases when analyzing slopes.
• The order in which vertices are connected matters.
• Not every polygon belongs to a special category.
• Coordinate geometry can also be used to find missing vertices, perimeter, and area.
• Good classification uses the minimum calculations necessary to establish the required properties.
• The final classification should be the most specific one supported by the coordinate evidence.

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Check Your Understanding

1. Explain how distance, slope, and midpoint can each provide different information about a polygon.
2. Triangle ABC has vertices A(0, 0), B(6, 0), and C(3, 4). Classify the triangle by side length.
3. Triangle ABC has vertices A(1, 1), B(6, 1), and C(6, 5). Classify the triangle by angle type and explain your reasoning.
4. Quadrilateral ABCD has vertices A(0, 0), B(6, 0), C(8, 4), and D(2, 4). Use slopes to determine whether it is a parallelogram.
5. Use the diagonal midpoints to verify your conclusion for the quadrilateral in Question 4.
6. Quadrilateral ABCD has vertices A(1, 1), B(7, 1), C(7, 5), and D(1, 5). Determine its most specific classification.
7. Quadrilateral ABCD has vertices A(0, 3), B(4, 0), C(8, 3), and D(4, 6). Use distance and slope to determine its most specific classification.
8. A quadrilateral has four congruent sides but no right angles. What is its most specific classification?
9. A parallelogram has one right angle. What more specific type of quadrilateral must it be? Explain.
10. A parallelogram has four congruent sides and one right angle. What is its most specific classification?
11. A quadrilateral's diagonals have the same midpoint. What does this tell you about the quadrilateral, assuming it is a simple quadrilateral?
12. A parallelogram has congruent diagonals. What more specific classification can be made?
13. Three vertices of parallelogram ABCD are A(2, 1), B(7, 1), and C(9, 5), listed consecutively. Determine the coordinates of D and explain the coordinate relationship you used.
14. A rectangular floor plan has vertices A(2, 2), B(10, 6), C(8, 10), and D(0, 6). Use coordinate geometry to verify that adjacent sides are perpendicular and opposite sides are parallel.
15. A student looks at a coordinate diagram and says, "It looks like a square, so it must be a square." Explain what calculations should be performed to verify the claim and why coordinate evidence is stronger than visual appearance.