Geometry on the Coordinate Plane

A coordinate plane allows us to describe geometric figures using the coordinates of their vertices.

For example, a rectangle might have vertices:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

From these coordinates, we can determine:

• side lengths
• perimeter
• area
• shape properties
• distances between vertices
• relationships between sides

This connects algebra and geometry.

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Reviewing Side Lengths

Before calculating perimeter or area, we often need to determine the lengths of the sides.

For horizontal sides:

length = |x₂ − x₁|

For vertical sides:

length = |y₂ − y₁|

For diagonal sides, use the distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Choosing the simplest appropriate method makes calculations faster.

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Horizontal Side Lengths

Suppose:

A(2, 3)

and:

B(9, 3)

Because the y-coordinates are the same, AB is horizontal.

Therefore:

AB = |9 − 2|

AB = 7 units

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Vertical Side Lengths

Suppose:

B(9, 3)

and:

C(9, 8)

Because the x-coordinates are the same, BC is vertical.

Therefore:

BC = |8 − 3|

BC = 5 units

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Diagonal Side Lengths

Suppose:

A(1, 2)

and:

B(5, 5)

Horizontal change:

5 − 1 = 4

Vertical change:

5 − 2 = 3

Therefore:

AB = √(4² + 3²)

AB = √25

AB = 5 units

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What Is Perimeter?

The perimeter of a polygon is the total distance around its outside boundary.

To calculate perimeter:

add the lengths of all sides

For a polygon with side lengths:

a, b, c, d

the perimeter is:

P = a + b + c + d

The exact formula depends on the shape, but the basic idea is always the same:

Perimeter = total boundary length

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Perimeter Units

Because perimeter measures length, it uses linear units.

Examples include:

• mm
• cm
• m
• km

If no measurement unit is given, write:

units

For example:

P = 24 units

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Perimeter of a Rectangle

Consider the rectangle:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

Find AB:

AB = 7 − 1 = 6

Find BC:

BC = 6 − 2 = 4

Opposite sides of a rectangle are equal, so:

CD = 6

DA = 4

Perimeter:

P = 6 + 4 + 6 + 4

P = 20 units

We could also use:

P = 2l + 2w

P = 2(6) + 2(4)

P = 20 units

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What Is Area?

Area measures the amount of two-dimensional space inside a figure.

Perimeter measures the:

boundary

Area measures the:

region inside the boundary

This distinction is very important.

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Area Units

Area uses square units.

Examples:

• mm²
• cm²
• m²
• km²

If no measurement unit is specified, write:

square units

For example:

A = 24 square units

or:

A = 24 units²

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Area of a Rectangle

The formula for the area of a rectangle is:

A = length × width

For our previous rectangle:

Length:

6 units

Width:

4 units

Therefore:

A = 6 × 4

A = 24 units²

Notice:

Perimeter = 20 units

but:

Area = 24 units²

They measure different things.

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Perimeter vs Area

A common mistake is confusing perimeter and area.

Perimeter

asks:

"How far is it around the figure?"

Area

asks:

"How much space is inside the figure?"

Think of:

Perimeter → fence

Area → floor

If you were fencing a field, you would need its perimeter.

If you were covering the field with grass, you would need its area.

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A Reliable Coordinate Perimeter Method

When finding the perimeter of a coordinate figure:

Step 1: Identify the vertices in order.

Step 2: Determine which points are connected.

Step 3: Calculate every side length.

Step 4: Use simple subtraction for horizontal or vertical sides.

Step 5: Use the distance formula for diagonal sides.

Step 6: Add all side lengths.

Step 7: Include the correct units.

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Worked Example 1: Rectangle

A rectangle has vertices:

A(2, 1)

B(10, 1)

C(10, 6)

D(2, 6)

Length:

10 − 2 = 8

Width:

6 − 1 = 5

Perimeter:

P = 2(8) + 2(5)

P = 16 + 10

P = 26 units

Area:

A = 8 × 5

A = 40 units²

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Worked Example 2: Square

A square has vertices:

A(−2, 1)

B(4, 1)

C(4, 7)

D(−2, 7)

Side length:

4 − (−2) = 6

Therefore:

P = 4(6)

P = 24 units

Area:

A = 6²

A = 36 units²

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Area of a Triangle

The area of a triangle is:

A = ½bh

where:

• b = base
• h = perpendicular height

The height must meet the base at a:

90° angle

It is not necessarily one of the triangle's sides.

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Triangle on a Coordinate Plane

Consider:

A(1, 2)

B(7, 2)

C(4, 6)

AB is horizontal.

Base:

7 − 1 = 6

The vertical distance from C to the line AB is:

6 − 2 = 4

Therefore:

A = ½(6)(4)

A = 12 units²

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Perimeter of the Same Triangle

For:

A(1, 2)

B(7, 2)

C(4, 6)

We already know:

AB = 6

Now find AC:

Horizontal change:

4 − 1 = 3

Vertical change:

6 − 2 = 4

Therefore:

AC = √(3² + 4²)

AC = 5

Similarly:

BC = 5

So:

P = 6 + 5 + 5

P = 16 units

The triangle has:

Perimeter = 16 units

Area = 12 units²

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Area Does Not Require Every Side Length

Notice that we needed all three side lengths to calculate the triangle's perimeter.

But for its area, we only needed:

• base
• perpendicular height

This is an important problem-solving idea.

Do not calculate information that you do not need.

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Right Triangles

Coordinate planes make the area of some right triangles especially easy to calculate.

Suppose:

A(2, 2)

B(8, 2)

C(2, 7)

AB is horizontal:

AB = 6

AC is vertical:

AC = 5

These sides are perpendicular.

Therefore:

A = ½(6)(5)

A = 15 units²

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Perimeter of a Right Triangle

Using the same triangle:

AB = 6

AC = 5

Find BC:

BC = √(6² + 5²)

BC = √61

Therefore:

P = 6 + 5 + √61

Exact perimeter:

P = 11 + √61 units

Since:

√61 ≈ 7.81

Approximate perimeter:

P ≈ 18.81 units

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Area of a Parallelogram

The area of a parallelogram is:

A = bh

where:

• b = base
• h = perpendicular height

The slanted side is not the height unless it is perpendicular to the base.

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Worked Example 3: Parallelogram

Consider:

A(1, 2)

B(7, 2)

C(9, 6)

D(3, 6)

Base AB:

7 − 1 = 6

Vertical height:

6 − 2 = 4

Therefore:

A = 6 × 4

A = 24 units²

Notice that the slanted side AD is not used as the height.

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Perimeter of the Parallelogram

AB:

6

AD has:

Horizontal change:

3 − 1 = 2

Vertical change:

6 − 2 = 4

Therefore:

AD = √(2² + 4²)

AD = √20

AD = 2√5

Opposite sides are equal.

Therefore:

P = 2(6) + 2(2√5)

P = 12 + 4√5 units

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Area of a Trapezoid

The area of a trapezoid is:

A = ½(b₁ + b₂)h

where:

• b₁ and b₂ are the parallel sides
• h is the perpendicular distance between them

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Worked Example 4: Trapezoid

Suppose the vertices are:

A(1, 2)

B(9, 2)

C(7, 6)

D(3, 6)

The parallel bases are:

AB = 8

and:

CD = 4

Height:

6 − 2 = 4

Therefore:

A = ½(8 + 4)(4)

A = ½(12)(4)

A = 24 units²

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Finding the Trapezoid's Perimeter

Using the same trapezoid:

AB = 8

CD = 4

For BC:

Horizontal change:

9 − 7 = 2

Vertical change:

6 − 2 = 4

So:

BC = √(2² + 4²)

BC = √20

BC = 2√5

Similarly:

DA = 2√5

Therefore:

P = 8 + 4 + 2√5 + 2√5

P = 12 + 4√5 units

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Irregular Coordinate Figures

Not every coordinate polygon is a simple rectangle, triangle, or trapezoid.

For an irregular figure, we can often divide it into simpler shapes.

For example, an irregular polygon might be divided into:

• rectangles
• triangles
• trapezoids

Calculate the area of each part and then add the results.

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Composite Figures

A composite figure is made from two or more simpler shapes.

Suppose an L-shaped figure can be divided into two rectangles.

Rectangle 1:

8 × 3 = 24

Rectangle 2:

4 × 2 = 8

Total area:

24 + 8 = 32 units²

Another method is to calculate the area of a large rectangle and subtract a missing rectangle.

Both methods should give the same result.

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Subtraction Method

Suppose a shape fits inside a:

10 × 8 rectangle

but has a:

4 × 3 rectangle

removed.

Large rectangle:

10 × 8 = 80

Missing rectangle:

4 × 3 = 12

Area of shape:

80 − 12 = 68 units²

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Be Careful with Composite Perimeters

Area and perimeter behave differently when shapes are combined.

For area:

add the areas of the pieces

But for perimeter, only add the lengths on the outside boundary.

Do not include interior lines used to divide the figure.

This is a common source of mistakes.

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Using Coordinates to Identify Shapes

Coordinates can tell us more than perimeter and area.

We can calculate side lengths to investigate whether a figure might be:

• square
• rectangle
• rhombus
• parallelogram
• isosceles triangle
• equilateral triangle

For example, if all four sides have equal length, the figure has an important property of a rhombus.

However, side lengths alone may not always be enough to completely classify the figure.

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Worked Example 5: Analyze a Quadrilateral

Consider:

A(0, 0)

B(6, 0)

C(6, 4)

D(0, 4)

Side lengths:

AB = 6

BC = 4

CD = 6

DA = 4

The sides are horizontal and vertical, so adjacent sides meet at right angles.

Therefore, the figure is a:

rectangle

Perimeter:

P = 6 + 4 + 6 + 4

P = 20 units

Area:

A = 6 × 4

A = 24 units²

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Worked Example 6: Square or Rectangle?

Consider:

A(−2, −2)

B(3, −2)

C(3, 3)

D(−2, 3)

Each side has length:

5 units

The sides are horizontal and vertical, so adjacent sides are perpendicular.

Therefore, the figure is a:

square

Perimeter:

P = 4(5) = 20 units

Area:

A = 5² = 25 units²

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Rotated Figures

A shape does not have to have horizontal and vertical sides.

Consider a square that has been rotated on the coordinate plane.

Its side lengths may need to be found using the distance formula.

We cannot assume that:

horizontal difference = side length

when the side is diagonal.

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Worked Example 7: Rotated Square

Consider the vertices:

A(0, 3)

B(3, 6)

C(6, 3)

D(3, 0)

Find AB:

AB = √[(3 − 0)² + (6 − 3)²]

AB = √(9 + 9)

AB = √18

AB = 3√2

All four sides have this length.

Therefore:

P = 4(3√2)

P = 12√2 units

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Area of the Rotated Square

The diagonals are easy to measure.

AC extends from:

(0, 3) to (6, 3)

So:

AC = 6

BD extends from:

(3, 6) to (3, 0)

So:

BD = 6

For this square, the area can be calculated from the diagonals:

A = ½d₁d₂

Therefore:

A = ½(6)(6)

A = 18 units²

This agrees with using:

side² = (3√2)² = 18

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Coordinate Area by Counting Squares

When a figure lies on a coordinate grid, we can sometimes estimate or calculate area by counting unit squares.

A single:

1 × 1

square has area:

1 square unit

This method is particularly useful for:

• visualizing area
• checking calculations
• simple rectangles
• simple composite figures

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Area by Enclosing Rectangle

Another useful method for irregular figures is to draw the smallest convenient rectangle around the polygon.

Then:

Area of polygon = Area of enclosing rectangle − Area outside polygon

The outside regions can often be divided into triangles or rectangles.

This method is particularly useful for slanted coordinate figures.

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Worked Example 8: Enclosing Rectangle

Suppose a triangle lies inside a:

8 × 6

rectangle.

Area of rectangle:

8 × 6 = 48

Suppose the three regions outside the triangle have areas:

6

10

and:

8

Then:

Triangle area = 48 − 6 − 10 − 8

Triangle area = 24 units²

This approach can sometimes be easier than finding a perpendicular height directly.

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Coordinate Geometry and Scale

Coordinate diagrams often represent real objects using a scale.

Suppose:

1 coordinate unit = 10 m

A rectangular property has coordinate dimensions:

8 units × 5 units

Actual dimensions:

80 m × 50 m

Perimeter:

P = 2(80) + 2(50)

P = 260 m

Area:

A = 80 × 50

A = 4000 m²

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Scale and Area

Be especially careful when converting area.

If:

1 coordinate unit = 5 m

then one square coordinate unit represents:

5 m × 5 m

which equals:

25 m²

Therefore:

1 unit² = 25 m²

not:

5 m²

This is because area involves two dimensions.

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Worked Example 9: Scale and Area

A coordinate rectangle measures:

6 units × 4 units

and:

1 unit = 3 m

Actual dimensions:

18 m × 12 m

Area:

A = 18 × 12

A = 216 m²

Alternatively:

Coordinate area:

6 × 4 = 24 units²

Since:

1 unit² = 3² = 9 m²

then:

24 × 9 = 216 m²

Both methods give the same answer.

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Real-World Application: Fencing

Suppose a rectangular garden has coordinate vertices:

(1, 2)

(11, 2)

(11, 8)

(1, 8)

If each coordinate unit represents:

1 metre

then:

Length:

10 m

Width:

6 m

Perimeter:

P = 2(10) + 2(6)

P = 32 m

The garden requires:

32 m of fencing

if the entire boundary is fenced.

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Real-World Application: Grass

Using the same garden:

Length:

10 m

Width:

6 m

Area:

A = 10 × 6

A = 60 m²

Therefore:

• fencing depends on perimeter
• grass coverage depends on area

This demonstrates why distinguishing the two quantities matters.

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Real-World Application: Flooring

Suppose a room is represented by a rectangle with dimensions:

8 m × 5 m

Flooring covers the interior.

Therefore, we need:

area

A = 8 × 5

A = 40 m²

If trim is placed around the edges of the room, we need:

perimeter

P = 2(8) + 2(5)

P = 26 m

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Real-World Application: Land Surveying

Coordinate geometry can be used to represent property boundaries.

Surveyors may work with coordinates that identify corners of a parcel of land.

From these coordinates, mathematical methods can help determine:

• boundary lengths
• perimeter
• area
• distances between points

For irregular parcels, more advanced coordinate-area methods may be used.

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Real-World Application: Computer Graphics

Digital shapes can be represented by vertices stored as coordinates.

Software can use these coordinates to calculate:

• edge lengths
• perimeter
• area
• centres
• collisions
• dimensions

Coordinate geometry therefore has important applications in:

• computer graphics
• game development
• CAD
• engineering
• digital mapping

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Real-World Application: Architecture

Architectural drawings often use scaled coordinate-like systems.

A floor plan may represent:

• walls
• rooms
• doors
• windows
• floor areas

Perimeter calculations can help determine quantities such as:

• trim
• fencing
• wall boundaries

Area calculations can help determine quantities such as:

• flooring
• tiles
• carpet
• paint coverage for flat surfaces
• usable floor space

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Worked Example 10: Practical Coordinate Problem

A rectangular playground has vertices:

A(2, 3)

B(14, 3)

C(14, 11)

D(2, 11)

Each coordinate unit represents:

2 m

Coordinate length:

14 − 2 = 12 units

Actual length:

12 × 2 = 24 m

Coordinate width:

11 − 3 = 8 units

Actual width:

8 × 2 = 16 m

Perimeter:

P = 2(24) + 2(16)

P = 80 m

Area:

A = 24 × 16

A = 384 m²

Therefore:

• 80 m of boundary surrounds the playground.
• The playground covers 384 m².

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Perimeter of Irregular Polygons

For an irregular polygon, calculate every outside side individually.

Suppose a pentagon has side lengths:

4

5

√13

7

and:

6

Then:

P = 4 + 5 + √13 + 7 + 6

P = 22 + √13 units

If a decimal is required:

√13 ≈ 3.61

Therefore:

P ≈ 25.61 units

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Keep Exact Values When Possible

If a diagonal side has length:

√20

do not immediately round it.

Simplify:

√20 = √(4 × 5)

√20 = 2√5

Keeping exact values prevents rounding errors.

Only convert to a decimal when required.

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Using Coordinates to Check a Shape

Suppose a quadrilateral appears to be a square.

A coordinate analysis could include:

Step 1: Calculate all four side lengths.

Step 2: Check whether they are equal.

Step 3: Check whether adjacent sides are perpendicular, if required.

Step 4: Calculate the perimeter.

Step 5: Calculate the area.

This is more reliable than judging the figure by appearance.

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Choosing the Correct Area Formula

Before calculating area, identify the shape.

Rectangle

A = lw

Square

A = s²

Triangle

A = ½bh

Parallelogram

A = bh

Trapezoid

A = ½(b₁ + b₂)h

For composite or irregular figures:

divide the figure into simpler shapes

or:

subtract unwanted areas from a larger shape

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A Reliable Area Strategy

Step 1: Plot or inspect the vertices.

Step 2: Identify the shape or divide it into simpler shapes.

Step 3: Determine the required dimensions from the coordinates.

Step 4: Choose the appropriate area formula.

Step 5: Calculate carefully.

Step 6: Use square units.

Step 7: Check whether the result is reasonable.

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Checking Perimeter Answers

Ask:

Did I include every outside side?

Did I accidentally include a diagonal that is not part of the boundary?

Did I use the distance formula for diagonal sides?

Did I use linear units?

These questions catch many common mistakes.

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Checking Area Answers

Ask:

Did I identify the correct shape?

Did I use perpendicular height rather than a slanted side?

Did I include all parts of a composite figure?

Did I accidentally count overlapping regions twice?

Did I use square units?

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Common Mistakes

Mistake 1: Confusing perimeter and area

Perimeter measures the boundary.

Area measures the interior.

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Mistake 2: Using square units for perimeter

Perimeter:

m

Area:

m²

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Mistake 3: Using linear units for area

An area of:

24 m

is incorrectly labeled.

It should be:

24 m²

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Mistake 4: Adding coordinates instead of finding side lengths

Coordinates represent positions.

You must determine the distance between the positions.

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Mistake 5: Using simple subtraction for a diagonal side

For diagonal segments, use:

distance formula

unless the length is otherwise known.

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Mistake 6: Using a slanted side as triangle height

The height must be:

perpendicular to the base

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Mistake 7: Using a parallelogram's slanted side as its height

Again, the height must be perpendicular to the base.

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Mistake 8: Including interior lines in a composite perimeter

Only count the:

outside boundary

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Mistake 9: Forgetting to square the scale factor for area

If:

1 unit = 4 m

then:

1 unit² = 16 m²

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Mistake 10: Rounding side lengths too early

Keep exact radical values until the final step whenever possible.

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Did You Know?

Coordinate geometry makes it possible for computers to calculate the area and perimeter of complicated shapes.

Modern applications include:

• geographic information systems (GIS)
• digital maps
• architecture
• land surveying
• computer-aided design
• robotics
• engineering
• computer graphics
• game development

A complicated boundary can be stored as a series of coordinate points.

Mathematical algorithms can then calculate its dimensions.

The same coordinate principles used in school geometry form the foundation for much more advanced applications.

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Key Terms

• Coordinate plane: Two-dimensional system used to locate points.
• Vertex: Point where two sides of a polygon meet.
• Polygon: Closed two-dimensional figure made from straight line segments.
• Side length: Distance between adjacent vertices.
• Perimeter: Total distance around the boundary of a figure.
• Area: Amount of two-dimensional space inside a figure.
• Distance formula: Formula used to calculate distance between coordinate points.
• Base: Chosen side of a figure used in an area calculation.
• Height: Perpendicular distance from a base to the opposite side or vertex.
• Composite figure: Figure made from two or more simpler shapes.
• Scale: Relationship between measurements on a representation and actual measurements.
• Square unit: Unit used for area.
• Diagonal: Segment joining two non-adjacent vertices of a polygon.
• Exact value: Value that has not been rounded.
• Approximation: Rounded value close to the exact value.

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Key Formulas

Horizontal length:

|x₂ − x₁|

Vertical length:

|y₂ − y₁|

Distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Rectangle perimeter:

P = 2l + 2w

Rectangle area:

A = lw

Square perimeter:

P = 4s

Square area:

A = s²

Triangle area:

A = ½bh

Parallelogram area:

A = bh

Trapezoid area:

A = ½(b₁ + b₂)h

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Key Takeaways

• Coordinates can be used to determine the dimensions of geometric figures.
• Horizontal side lengths are found using differences between x-coordinates.
• Vertical side lengths are found using differences between y-coordinates.
• Diagonal side lengths can be calculated using the distance formula.
• Perimeter is the total distance around a polygon.
• Perimeter uses linear units such as cm, m, or km.
• Area measures the amount of space inside a two-dimensional figure.
• Area uses square units such as cm², m², or km².
• Perimeter and area measure different properties and should not be confused.
• Rectangles and squares can often be analyzed directly from coordinate differences.
• Triangle area requires a base and perpendicular height.
• Parallelogram area also uses perpendicular height rather than the length of a slanted side.
• Trapezoid area depends on the two parallel bases and their perpendicular separation.
• Irregular coordinate figures can often be divided into simpler shapes.
• An enclosing rectangle can sometimes be used to calculate an irregular area by subtraction.
• Composite perimeters include only the outside boundary, not interior dividing lines.
• Exact radical values should usually be kept until the final stage of a calculation.
• Coordinate calculations can also help identify and analyze geometric shapes.
• Scale factors must be interpreted carefully: if lengths are multiplied by a scale factor, areas are multiplied by the square of that scale factor.
• Coordinate perimeter and area calculations have practical applications in construction, architecture, mapping, surveying, landscaping, computer graphics, engineering, and land measurement.