Understanding Area and Perimeter in the Coordinate Plane

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.

	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	$ℓ \approx 8.5$	$w \approx 5.4$

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

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

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

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
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 $K (4, - 2)$	$M K = (4 - (- 2))^{2} + (- 2 - (- 2))^{2}$	$M K = 6$

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

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
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 $(7, 1)$	$c = (7 - 2)^{2} + (1 - 3)^{2}$	$c \approx 5.4$

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

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
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 $C (8, 0)$	$B C = (8 - 4)^{2} + (0 - 7)^{2}$	$B C \approx 8.1$

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

In the diagram three segments are defined.

Diagram with three segments drawn in it.

What is the length of

\overline{A B} ?

Answer in exact form.

What is the length of

\overline{C D} ?

Answer in exact form.

What is the length of

\overline{E F} ?

To find the length of a segment, we can use the Distance Formula. First, we need to identify the coordinates of the segment's endpoints, A and B.

The segment has its endpoints at ( -3, 1) and ( 7, 6). Let's substitute these ordered pairs into the Distance Formula and simplify the expression.

The segment has a length of 5sqrt(5) units.

Let's identify the coordinates of the endpoints of CD.

This segment has its endpoints at ( - 7, - 8) and ( - 3, - 4). By using these points in the Distance Formula we find the length of the segment.

The segment's length is 4sqrt(2) units.

Here we need to find where the segment EF has its endpoints.

We can see that it has its endpoints at ( 3, - 2) and ( 6, -6). Let's substitute these points into the Distance Formula.

The length of the segment is 5 units.

A polygon has three vertices.

U (- 2, 4) V (3, 4) W (3, - 4)

Find the polygon's perimeter and round the answer to two decimal places.

Find the polygon's area.

To determine the perimeter of the polygon, we need to find the sum of its side lengths. Let's draw it in a coordinate plane.

From the diagram we can tell that it is a right triangle. Before finding the sum of the side lengths, we must find the length of each side.

Length of WU

To find the length of WU, we will substitute the points ( - 2, 4) and ( 3, - 4) into the Distance Formula.

Remaining lengths

We continue by calculating the length of the other two sides UV and VW. Since UV is a vertical segment and VW is a vertical segment we can use the Ruler Postulate.

Side	Coordinates	Ruler Postulate	Length
UV	( - 2,4) ( 3,4)	\| 3-( - 2)\|	5
VW	(3, 4) (3, - 4)	\| - 4- 4\|	8

Perimeter

Let's add the side lengths to find the triangle's perimeter.

The triangle's perimeter is approximately 22.43 units.

The area of a triangle is half the product of the base and the height. A=1/2 b h We can use the triangle's vertical side as the base and its horizontal side as the height.

In Part A we calculated the lengths of these two sides. base= 8 height= 5 Let's calculate the triangle's area.

The area is 20 units^2.

Consider the following parallelogram.

Diagram with a parallelogram with vertices in (-5,-2), (-3,3), (4,3), and (2,-2).

Find the perimeter of the parallelogram.

Find the area of the parallelogram.

To determine the perimeter, we need to find the length of each side.

Side Lengths

To calculate the lengths of the sides, we first need to identify the vertices of the parallelogram.

We find the length of AB by substituting its endpoints into the Distance Formula.

The side has a length of 5 units. As for parallelograms, opposite sides are congruent. Therefore, CD must also be 5 units long. To find the lengths of the sides that are parallel with the x-axis, we will use the Ruler Postulate.

Side	Coordinates	Ruler Postulate	Length
BC	( - 2,2) ( 5,2)	\| 5-( - 2)\|	7
AD	( - 5,- 2) ( 2,- 2)	\| 2-( - 5)\|	7

Calculating the Perimeter

Let's add the side lengths to find the parallelogram's perimeter.

The perimeter is 24 units long.

To determine the area of the parallelogram, we will need to find the length of its base and height. The height of the parallelogram is perpendicular to its base. Let's draw it.

We can calculate the height h of the parallelogram using the Ruler Postulate and the y-coordinates of its endpoints.

The area of a parallelogram is the product of its base and its height. Let's substitute the corresponding values into the formula.

The parallelogram's area is 28 units^2.

Find the perimeter of the compound figure. Round your answer to two decimals.

We will first calculate the length of each side of the compound figure. Then, we will find the perimeter by adding the side lengths.

Calculating Side Lengths

Let's identify the coordinates of each vertex.

We find the length of AB by substituting the points ( 0, - 2) and ( - 3, 1) into the Distance Formula.

The side has a length of sqrt(18) units. We calculate the lengths of the other four sides using the same method.

Side	Points	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
BC	( - 3, 1) ( 2, 6)	sqrt(( 2-( - 3))^2+( 6- 1)^2)	sqrt(50)
CD	( 2, 6) ( 7, 1)	sqrt(( 7- 2)^2+( 1- 6)^2)	sqrt(50)
DE	( 7, 1) ( 3, 1)	sqrt(( 3- 7)^2+( 1- 1)^2)	4
AE	( 0, - 2) ( 3, 1)	sqrt(( 3- 0)^2+( 1-( - 2))^2)	sqrt(18)

Perimeter

Let's add the side lengths to find the perimeter of the compound figure.

The perimeter is approximately 26.63 units long.

The points

A,

B,

and

C

define a triangle.

A (2, 4) B (6, 9) C (8, 3)

△ A B C

equilateral, isosceles, or scalene?

We have been given the coordinates of the vertices of △ ABC. A(2,4) B(6,9) C(8,3) We want to know if the triangle is equilateral, isosceles, or scalene. Let's recall the relationship between the sides for each one of these.

Type of Triangle	Relationship Between Sides
Equilateral	All sides are congruent.
Isosceles	Exactly two sides are congruent.
Scalene	All sides have different lengths.

Let's draw the triangle in a diagram.

Unfortunately, we cannot determine any side length directly from this image. Therefore, we need to calculate the length of each side.

Calculating Side Lengths

We will calculate the side lengths using the Distance Formula. Let's start with AB.

The side AB has the length sqrt(41) units. We will now repeat this for the other two sides.

Side	Points	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
BC	( 6, 9) ( 8, 3)	sqrt(( 8- 6)^2+( 3- 9)^2)	sqrt(40)
AC	( 2, 4) ( 8, 3)	sqrt(( 8- 2)^2+( 3- 4)^2)	sqrt(37)

Determining the Triangle's Type

Let's summarize the length of the sides. AB = sqrt(41) BC = sqrt(40) AC = sqrt(37) Since no sides are congruent, it is a scalene triangle.

A quadrilateral has four vertices.

A (- 3, - 2) B (3, 3) C (5, 2) D (- 1, - 3)

Is the quadrilateral a parallelogram or a trapezoid?

We want to know if a quadrilateral is a parallelogram or a trapezoid. Let's draw it in a diagram.

In a parallelogram, opposite sides are congruent. To determine if this is true for our quadrilateral, we need to find the length of each side.

Calculating Side Lengths

We find the length of AB by substituting the points ( - 3, - 2) and ( 3, 3) into the Distance Formula.

The side has a length of sqrt(61) units. We calculate the lengths of the other three sides using the same method.

Side	Points	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
BC	( 3, 3) ( 5, 2)	sqrt(( 5- 3)^2+( 2- 3)^2)	sqrt(5)
CD	( 5, 2) ( - 1, - 3)	sqrt(( - 1- 5)^2+( - 3- 2)^2)	sqrt(61)
AD	( - 3, - 2) ( - 1, - 3)	sqrt(( - 1-( - 3))^2+( - 3-( - 2))^2)	sqrt(5)

Comparing Side Lengths

We have found the length of each of the quadrilateral's four sides. Let's investigate if opposite side are congruent.

Side	Length
AB	AB= sqrt(61)
BC	BC= sqrt(5)
CD	CD= sqrt(61)
AD	AD= sqrt(5)

Since opposite sides have the same length, they are congruent. Therefore, the quadrilateral is a parallelogram.

Perimeter and Area in a Coordinate Plane

Catch-Up and Review

Investigate Shapes by Using Coordinates

Solving Problems in the Real World by Using Coordinates

Hint

Solution

Investigate the Properties of a Triangle

Hint

Solution

Investigate the Properties of a Compound Shape

Hint

Solution

Square Room

Rectangular Room

Triangular Room

Total Area

Investigate the Properties of a Triangle

Hint

Solution

Classifying Triangles by Using Coordinates

Hint

Solution

Length of WU

Remaining lengths

Perimeter

Side Lengths

Calculating the Perimeter

Calculating Side Lengths

Perimeter

Calculating Side Lengths

Determining the Triangle's Type

Calculating Side Lengths

Comparing Side Lengths

Perimeter and Area in a Coordinate Plane

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