Finding the Length of the Sides

Oct 13, 2025 by windailys.com 90 views

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Introduction

In geometry, a rectangle is a type of quadrilateral with four right angles and opposite sides of equal length. Given the coordinates of its vertices, we can calculate the perimeter and area of the rectangle. In this article, we will find the perimeter and area of a rectangle with vertices ${(-4, 4), (0, 4), (0, -1),}$ and ${(-4, -1)}$ .

Understanding the Problem

To find the perimeter and area of the rectangle, we need to understand the concept of distance between two points in a coordinate plane. The distance between two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ is given by the formula:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

We will use this formula to find the length of the sides of the rectangle.

Finding the Length of the Sides

Let's find the length of the sides of the rectangle using the distance formula.

Side 1: From (-4, 4) to (0, 4)

Using the distance formula, we get:

${d_1 = \sqrt{(0 - (-4))^2 + (4 - 4)^2} = \sqrt{16 + 0} = \sqrt{16} = 4}$

So, the length of side 1 is 4 units.

Side 2: From (0, 4) to (0, -1)

Using the distance formula, we get:

${d_2 = \sqrt{(0 - 0)^2 + (-1 - 4)^2} = \sqrt{0 + 25} = \sqrt{25} = 5}$

So, the length of side 2 is 5 units.

Side 3: From (0, -1) to (-4, -1)

Using the distance formula, we get:

${d_3 = \sqrt{(-4 - 0)^2 + (-1 - (-1))^2} = \sqrt{16 + 0} = \sqrt{16} = 4}$

So, the length of side 3 is 4 units.

Side 4: From (-4, -1) to (-4, 4)

Using the distance formula, we get:

${d_4 = \sqrt{(-4 - (-4))^2 + (4 - (-1))^2} = \sqrt{0 + 25} = \sqrt{25} = 5}$

So, the length of side 4 is 5 units.

Calculating the Perimeter

The perimeter of a rectangle is the sum of the lengths of its four sides. In this case, the perimeter is:

${P = d_1 + d_2 + d_3 + d_4 = 4 + 5 + 4 + 5 = 18}$

So, the perimeter of the rectangle is 18 units.

Calculating the Area

The area of a rectangle is the product of its length and width. In this case, the length is 5 units and the width is 4 units. Therefore, the area is:

${A = \text{length} \times \text{width} = 5 \times 4 = 20}$

So, the area of the rectangle is 20 square units.

Conclusion

In this article, we found the perimeter and area of a rectangle with vertices ${(-4, 4), (0, 4), (0, -1),}$ and ${(-4, -1)}$ . We used the distance formula to find the length of the sides of the rectangle and then calculated the perimeter and area. The perimeter of the rectangle is 18 units and the area is 20 square units.

References

[1] Khan Academy. (n.d.). Distance Formula. Retrieved from https://www.khanacademy.org/math/geometry/geometry-distance-formula
[2] Math Open Reference. (n.d.). Rectangle. Retrieved from https://www.mathopenref.com/rectangle.html

Future Work

In the future, we can explore other types of quadrilaterals and calculate their perimeter and area using the distance formula. We can also investigate the properties of rectangles and other types of quadrilaterals.

Code

Here is some sample code in Python to calculate the perimeter and area of a rectangle:

import math

def distance(x1, y1, x2, y2):
    return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

def perimeter(a, b):
    return 2 * (a + b)

def area(a, b):
    return a * b

# Define the coordinates of the vertices
x1, y1 = -4, 4
x2, y2 = 0, 4
x3, y3 = 0, -1
x4, y4 = -4, -1

# Calculate the length of the sides
d1 = distance(x1, y1, x2, y2)
d2 = distance(x2, y2, x3, y3)
d3 = distance(x3, y3, x4, y4)
d4 = distance(x4, y4, x1, y1)

# Calculate the perimeter and area
p = perimeter(d1, d2)
a = area(d1, d2)

print("Perimeter:", p)
print("Area:", a)

This code defines a function distance to calculate the distance between two points, and then uses this function to calculate the length of the sides of the rectangle. It then calculates the perimeter and area using the perimeter and area functions, respectively.

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Introduction

In our previous article, we discussed how to find the perimeter and area of a rectangle with given vertices. In this article, we will answer some frequently asked questions related to rectangle perimeter and area.

Q: What is the formula for the perimeter of a rectangle?

A: The formula for the perimeter of a rectangle is:

${P = 2 \times (\text{length} + \text{width})}$

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is:

${A = \text{length} \times \text{width}}$

Q: How do I find the length and width of a rectangle?

A: To find the length and width of a rectangle, you need to know the coordinates of its vertices. You can use the distance formula to find the length and width.

Q: What is the distance formula?

A: The distance formula is:

${d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

This formula calculates the distance between two points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ in a coordinate plane.

Q: How do I use the distance formula to find the length and width of a rectangle?

A: To use the distance formula to find the length and width of a rectangle, you need to know the coordinates of its vertices. For example, if the vertices are ${(x_1, y_1)}$ , ${(x_2, y_2)}$ , ${(x_3, y_3)}$ , and ${(x_4, y_4)}$ , you can use the distance formula to find the length and width as follows:

Length: ${d_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$
Width: ${d_2 = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}}$

Q: What is the difference between the perimeter and area of a rectangle?

A: The perimeter of a rectangle is the distance around its edges, while the area is the amount of space inside the rectangle.

Q: How do I calculate the perimeter and area of a rectangle with given vertices?

A: To calculate the perimeter and area of a rectangle with given vertices, you can use the formulas:

Perimeter: ${P = 2 \times (\text{length} + \text{width})}$
Area: ${A = \text{length} \times \text{width}}$

You can also use the distance formula to find the length and width of the rectangle, and then use these values to calculate the perimeter and area.

Q: What are some real-world applications of rectangle perimeter and area?

A: Rectangle perimeter and area have many real-world applications, such as:

Building design: Architects use rectangle perimeter and area to design buildings and ensure that they have enough space for the occupants.
Landscaping: Gardeners use rectangle perimeter and area to design gardens and ensure that they have enough space for the plants.
Packaging: Manufacturers use rectangle perimeter and area to design packaging and ensure that it is the right size for the product.

Q: How do I use Python to calculate the perimeter and area of a rectangle?

A: You can use the following Python code to calculate the perimeter and area of a rectangle:

import math

def distance(x1, y1, x2, y2):
    return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

def perimeter(a, b):
    return 2 * (a + b)

def area(a, b):
    return a * b

# Define the coordinates of the vertices
x1, y1 = -4, 4
x2, y2 = 0, 4
x3, y3 = 0, -1
x4, y4 = -4, -1

# Calculate the length and width of the rectangle
d1 = distance(x1, y1, x2, y2)
d2 = distance(x2, y2, x3, y3)

# Calculate the perimeter and area of the rectangle
p = perimeter(d1, d2)
a = area(d1, d2)

print("Perimeter:", p)
print("Area:", a)

This code defines a function distance to calculate the distance between two points, and then uses this function to calculate the length and width of the rectangle. It then calculates the perimeter and area using the perimeter and area functions, respectively.

Conclusion

In this article, we answered some frequently asked questions related to rectangle perimeter and area. We discussed the formulas for the perimeter and area of a rectangle, and how to use the distance formula to find the length and width of a rectangle. We also provided some real-world applications of rectangle perimeter and area, and showed how to use Python to calculate the perimeter and area of a rectangle.