Understanding Area and Perimeter in the Coordinate Plane

On a coordinate plane, each point has coordinates. The coordinates of a line segment's endpoints can be used to calculate the line segment's length. In this lesson, the perimeter and area of different polygons will be calculated using the vertices' coordinates.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Right triangles
Pythagorean Theorem
Distance Formula
Perimeter
Area

Try your knowledge on these topics.

a State the coordinates of the points plotted on the plane.

b Determine whether a triangle with side lengths

4,

5,

and

6

is a right triangle.

c Use the Pythagorean Theorem to find the missing side length. Round the answer to one decimal place.

A right triangle with given side lengths

d Calculate

M N

by using the Distance Formula.

e Find the perimeter of the given polygon.

f Calculate the area of the rectangle.

A rectangle with the given width and length

Challenge

Investigate Shapes by Using Coordinates

The coordinates of the vertices of a polygon can be used to identify the type of the polygon.

Emily's sister completed her first $1 -$ mile run. Emily wants to make a congratulatory card in the shape of a triangle for her. She wants the card to have the form of an equilateral triangle. However, she lost her ruler and made all the measurements by eyeballing it.

How can Emily determine if the card has the shape of an equilateral, an isosceles, or a scalene triangle?

Example

Solving Problems in the Real World by Using Coordinates

Izabella wants to buy a plot of land for her garden. This rectangular plot of land should have a minimum area of

30

square meters and a perimeter no less than

20

meters to have enough space. She knows the coordinates of the vertices of the plot of land.

External credits: @kdekiara, www.textures.com

Does this plot of land satisfy Izabella's requirements?

Hint

Use the Distance Formula to find the width and the length of the rectangle.

Solution

The area and perimeter of a rectangle can be calculated using the following formulas.

Area : Perimeter : A = w ℓ P = 2 (w + ℓ)

Here,

w

and

ℓ

are the width and the length of the rectangle. In the diagram, the coordinates of the vertices are given. Using the Distance Formula, the length and width of a rectangle can be calculated.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

First, the

length

and

width

of the rectangle need to be identified.

External credits: @kdekiara, www.textures.com

Next, by substituting the endpoints of the chosen segments into the Distance Formula, their lengths can be calculated.

	Length	Width
Endpoints	$(- 5, 2)$ and $(3, 5)$	$(3, 5)$ and $(5, 0)$
Substitute	$ℓ = (3 - (- 5))^{2} + (5 - 2)^{2}$	$w = (5 - 3)^{2} + (0 - 5)^{2}$
Evaluate	$ℓ \approx 8.5$	$w \approx 5.4$

Now that the length and the width are known, the area and the perimeter of the rectangle can be determined.

	Area	Perimeter
Formula	$A = w ℓ$	$P = 2 (w + ℓ)$
Substitute	$A = 5.4 (8.5)$	$P = 2 (5.4 + 8.5)$
Evaluate	$A = 45.9$	$P = 27.8$

Since these values are greater than the area and the perimeter of the plot of land that Izabella wanted to buy, they satisfy the given requirements. Therefore, Izabella should buy this plot of land.

Example

Investigate the Properties of a Triangle

On a test, Kriz is asked to find the area and perimeter of a triangle illustrated on a coordinate plane.

Kriz wrote that the area of $△ M N K$ is $15$ square units and its perimeter is $24$ units. Is Kriz correct?

Hint

Find the coordinates of the vertices using the coordinate plane. Then use the Distance Formula to calculate the side lengths of the triangle.

Solution

First, the coordinates of the vertices on the coordinate plane need to be noted.

Next, the side lengths of

△ M N K

need to be calculated by using the Distance Formula. The calculations for

M N

are shown below.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

M N = (3 - (- 2))^{2} + (4 - (- 2))^{2}

Evaluate right-hand side

SubNeg

$a - (- b) = a + b$

M N = 5^{2} + 6^{2}

CalcPow

Calculate power

M N = 25 + 36

AddTerms

Add terms

M N = 61

CalcRoot

Calculate root

M N = 7.810249 \dots

RoundDec

Round to $1$ decimal place(s)

M N \approx 7.8

By following the same procedure,

N K

and

M K

can be calculated.

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Side	Endpoints	Substitute	Evaluate
$\overline{M N}$	$M (- 2, - 2)$ and $N (3, 4)$	$M N = (3 - (- 2))^{2} + (4 - (- 2))^{2}$	$M N \approx 7.8$
$\overline{N K}$	$N (3, 4)$ and $K (4, - 2)$	$N K = (4 - 3)^{2} + (- 2 - 4)^{2}$	$N K \approx 6.1$
$\overline{M K}$	$M (- 2, - 2)$ and $K (4, - 2)$	$M K = (4 - (- 2))^{2} + (- 2 - (- 2))^{2}$	$M K = 6$

Now, the perimeter of

△ M N K

can be calculated by adding the three side lengths of the triangle.

P = 7.8 + 6.1 + 6 ⇕ P = 19.9

The perimeter of

△ M N K

19.9

units. Next, its area will be found. Recall that the area of a triangle is half the product of the base and its corresponding height.

A = \frac{b h}{2}

In this formula,

b

is the base of the triangle and

h

its corresponding height. In a triangle, the base and its corresponding height are perpendicular. If

\overline{M K}

is the base, then the height is the perpendicular segment to

\overline{M K}

through

N .

In the diagram, it can be seen that the length of this segment is

6

units.

Finally, by substituting

b = 6

and

h = 6,

the area of

△ M N K

can be calculated.

A = \frac{b h}{2}

SubstituteII

$b = 6$ , $h = 6$

A = \frac{6 ( 6 )}{2}

Evaluate right-hand side

Multiply

A = \frac{3 6}{2}

CalcQuot

Calculate quotient

A = 18

Therefore, the area of the triangle is

18

square units and its perimeter

19.9

units. This means that Kriz was not correct.

Example

Investigate the Properties of a Compound Shape

The plan of a flat is represented on a coordinate plane. The plan has the shape of a compound geometric figure.

Calculate the area and the perimeter of the plan. Round each value to one decimal place.

Hint

Use the coordinate plane to find the side lengths of the compound figure. The area of a compound figure is equal to the sum of the areas of the geometric figures that make the compound figure.

Solution

Start by identifying the coordinates of the vertices and the lengths of the horizontal and vertical sides of the figure. Let $m$ be the length of the side which is neither vertical nor horizontal.

The value of

m

can be calculated by substituting

(- 5, 1)

and

(0, 4)

into the Distance Formula.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

m = (0 - (- 5))^{2} + (4 - 1)^{2}

Evaluate right-hand side

SubNeg

$a - (- b) = a + b$

m = 5^{2} + (4 - 1)^{2}

SubTerm

Subtract term

m = 5^{2} + 3^{2}

CalcPow

Calculate power

m = 25 + 9

AddTerms

Add terms

m = 34

CalcRoot

Calculate root

m = 5.830951 \dots

RoundDec

Round to $1$ decimal place(s)

m \approx 5.8

By adding the lengths of the exterior sides, the perimeter of the plan can be calculated.

P = 3 + 5.8 + 6 + 6 + 6 + 5 ⇕ P = 31.8

The perimeter of the figure is

31.8

units. The area of a compound figure is equal to the sum of the geometric figures' areas that make the compound figure. From the diagram, it can be seen that the flat has a square, a rectangular, and a triangular room.

Square Room

The area of a square is equal to the square of a side length. It has been previously found that, in the plan, each side of the square room is $6$ units long.

By raising

6

to the power of two, the area of the square can be determined.

A_{square} = 6^{2} ⇕ A_{square} = 36 units^{2}

Rectangular Room

The area of a rectangle is equal to the product of its length and width. From the diagram, it can be seen that the length of the rectangle is $5$ units and the width is $3$ units.

By multiplying

3

5,

the area of the rectangle can be found.

A_{rectangle} = 3 (5) ⇕ A_{rectangle} = 15 units^{2}

Triangular Room

The area of a triangle is half the product of its base and its height. Analyzing the diagram, it can be seen that the base and height of the triangle are $b = 5$ units and $h = 3$ units, respectively.

By substituting these values into the formula for the area of a triangle, its area can be calculated.

A = \frac{b h}{2}

SubstituteII

$b = 5$ , $h = 3$

A = \frac{5 ( 3 )}{2}

Evaluate right-hand side

Multiply

A = \frac{1 5}{2}

CalcQuot

Calculate quotient

A = 7.5

Total Area

Gather all the areas that have been found.

A_{square} A_{rectangle} A_{triangle} = 36 units^{2} = 15 units^{2} = 7.5 units^{2}

By adding these values, the area of the compound figure can be calculated.

A_{total} = 36 + 15 + 7.5 ⇕ A_{total} = 58.5 units^{2}

Example

Investigate the Properties of a Triangle

The diagram illustrates a triangle on a coordinate plane, and gives the coordinates of the vertices.

A triangle on a coordinate plane with the coordinates of the vertices

Calculate the area of the triangle. If needed, round the answer to one decimal place.

Hint

Use the formula $A = s (s - a) (s - b) (s - c),$ where $s$ is the semi-perimeter, and $a,$ $b,$ and $c$ are the side lengths of the triangle.

Solution

To calculate the area of a triangle, Heron's formula can be used.

A = s (s - a) (s - b) (s - c)

Here,

a,

b,

and

c

are the side lengths and

s

is the semi-perimeter of the triangle. The semi-perimeter is half the perimeter.

s = \frac{a + b + c}{2}

In the diagram, the coordinates of the vertices are given. The sides of the triangle will be labeled as

a,

b,

and

c .

By substituting the coordinates of the vertices into the Distance Formula, the side lengths can be calculated.

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Side	Endpoints	Substitute	Evaluate
$a$	$(2, 3)$ and $(3, 7.5)$	$a = (3 - 2)^{2} + (7.5 - 3)^{2}$	$a \approx 4.6$
$b$	$(3, 7.5)$ and $(7, 1)$	$b = (7 - 3)^{2} + (1 - 7.5)^{2}$	$b \approx 7.6$
$c$	$(2, 3)$ and $(7, 1)$	$c = (7 - 2)^{2} + (1 - 3)^{2}$	$c \approx 5.4$

Next, the side lengths will be substituted into the formula for the semi-perimeter to find the value of

s .

s = \frac{a + b + c}{2}

SubstituteValues

Substitute values

s = \frac{4 . 6 + 7 . 6 + 5 . 4}{2}

Evaluate right-hand side

AddTerms

Add terms

s = \frac{1 7 . 6}{2}

CalcQuot

Calculate quotient

s = 8.8

Now that the semi-perimeter of the triangle is known, its area can be found using Heron's Formula.

A = s (s - a) (s - b) (s - c)

SubstituteValues

Substitute values

A = 8.8 (8.8 - 4.6) (8.8 - 7.6) (8.8 - 5.4)

Evaluate right-hand side

SubTerms

Subtract terms

A = 8.8 (4.2) (1.2) (3.4)

Multiply

A = 150.7968

RoundDec

Round to $1$ decimal place(s)

A \approx 150.8

The area of the triangle is approximately

150.8

square units.

Closure

Classifying Triangles by Using Coordinates

With the topics seen in this lesson, the challenge presented at the beginning can be solved. It consisted in determining whether the card made by Emily has the shape of an equilateral, an isosceles, or a scalene triangle.

Hint

Plot the card on a coordinate plane so that one of its vertices is at the origin. Identify the coordinates of the vertices and then use them to calculate the side lengths.

Solution

The type of triangle can be identified by comparing the side lengths. In order to find them, the coordinates of the vertices should be known. Plot the card on a coordinate plane so that one of its vertices is at the origin.

Note that the bottom side is a horizontal segment. Therefore, its length is equal to the difference of the

x -

coordinates of the endpoints.

A (0, 0) and C (8, 0) ⇓ A C = 8 - 0 A C = 8 - 0

The lengths of the other two sides can be calculated using the Distance Formula.

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Side	Endpoints	Substitute	Evaluate
$\overline{A B}$	$A (0, 0)$ and $B (4, 7)$	$A B = (4 - (0))^{2} + (7 - 0)^{2}$	$A B \approx 8.1$
$\overline{B C}$	$B (4, 7)$ and $C (8, 0)$	$B C = (8 - 4)^{2} + (0 - 7)^{2}$	$B C \approx 8.1$

As it can be seen, $\overline{A B}$ and $\overline{B C}$ have the same length. However, they are a little longer than $\overline{A C} .$ Therefore, Emily's card has a shape of an isosceles triangle.

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Catch-Up and Review

Challenge

Investigate Shapes by Using Coordinates

Example

Solving Problems in the Real World by Using Coordinates

Hint

Solution

Example

Investigate the Properties of a Triangle

Hint

Solution

Example

Investigate the Properties of a Compound Shape

Hint

Solution

Square Room

Rectangular Room

Triangular Room

Total Area

Example

Investigate the Properties of a Triangle

Hint

Solution

Closure

Classifying Triangles by Using Coordinates

Hint

Solution

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