Compute The Area Of A Rectangle With Vertices At { (-3,-1), (1,3), (3,1),$}$ And { (-1,-3)$} . E N T E R Y O U R A N S W E R I N T H E B O X . D O N O T R O U N D A N Y S I D E L E N G T H S . .Enter Your Answer In The Box. Do Not Round Any Side Lengths. . E N T Eryo U R An S W Er In T H E B O X . Do N O T Ro U N D An Ys I D E L E N G T H S . { \square\$} Square Units

Introduction

In geometry, the area of a rectangle can be calculated using various methods, including the use of its vertices. Given the vertices of a rectangle, we can compute its area by finding the length of its sides and multiplying them together. In this article, we will explore how to compute the area of a rectangle with vertices at ${(-3,-1), (1,3), (3,1),}$ and ${(-1,-3)}$. We will use the distance formula to find the length of the sides and then multiply them together to get the area.

Understanding the Distance Formula

The distance formula is a fundamental concept in geometry that allows us to find the distance between two points in a coordinate plane. The formula is given by:

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

where ${(x_1, y_1)}$ and ${(x_2, y_2)}$ are the coordinates of the two points.

Finding the Length of the Sides

To find the length of the sides of the rectangle, we need to use the distance formula. Let's start by finding the length of the side between the points ${(-3,-1)}$ and ${(1,3)}$. We can plug these values into the distance formula:

${d_1 = \sqrt{(1 - (-3))^2 + (3 - (-1))^2}}$ ${d_1 = \sqrt{(4)^2 + (4)^2}}$ ${d_1 = \sqrt{16 + 16}}$ ${d_1 = \sqrt{32}}$

Similarly, we can find the length of the side between the points ${(1,3)}$ and ${(3,1)}$:

${d_2 = \sqrt{(3 - 1)^2 + (1 - 3)^2}}$ ${d_2 = \sqrt{(2)^2 + (-2)^2}}$ ${d_2 = \sqrt{4 + 4}}$ ${d_2 = \sqrt{8}}$

Next, we can find the length of the side between the points ${(3,1)}$ and ${(-1,-3)}$:

${d_3 = \sqrt{(-1 - 3)^2 + (-3 - 1)^2}}$ ${d_3 = \sqrt{(-4)^2 + (-4)^2}}$ ${d_3 = \sqrt{16 + 16}}$ ${d_3 = \sqrt{32}}$

Finally, we can find the length of the side between the points ${(-1,-3)}$ and ${(-3,-1)}$:

${d_4 = \sqrt{(-3 - (-1))^2 + (-1 - (-3))^2}}$ ${d_4 = \sqrt{(-2)^2 + (2)^2}}$ ${d_4 = \sqrt{4 + 4}}$ ${d_4 = \sqrt{8}}$

Computing the Area of the Rectangle

Now that we have found the length of the sides, we can compute the area of the rectangle by multiplying the length and width together. Since the length and width are the same, we can simply multiply the length by itself:

${Area = d_1 \times d_2}$ ${Area = \sqrt{32} \times \sqrt{8}}$ ${Area = \sqrt{32 \times 8}}$ ${Area = \sqrt{256}}$ ${Area = 16}$

Conclusion

In this article, we have shown how to compute the area of a rectangle with vertices at ${(-3,-1), (1,3), (3,1),}$ and ${(-1,-3)}$. We used the distance formula to find the length of the sides and then multiplied them together to get the area. The final answer is $\boxed{16}$ square units.

Introduction

In our previous article, we explored how to compute the area of a rectangle with vertices at ${(-3,-1), (1,3), (3,1),}$ and ${(-1,-3)}$. We used the distance formula to find the length of the sides and then multiplied them together to get the area. In this article, we will answer some frequently asked questions about computing the area of a rectangle with given vertices.

Q: What is the distance formula?

A: The distance formula is a fundamental concept in geometry that allows us to find the distance between two points in a coordinate plane. The formula is given by:

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

where ${(x_1, y_1)}$ and ${(x_2, y_2)}$ are the coordinates of the two points.

Q: How do I find the length of the sides of a rectangle with given vertices?

A: To find the length of the sides of a rectangle with given vertices, you can use the distance formula. Simply plug in the coordinates of the two points into the formula and simplify.

Q: What if the vertices of the rectangle are not given? Can I still compute the area?

A: Yes, you can still compute the area of a rectangle even if the vertices are not given. You can use the formula for the area of a rectangle, which is given by:

${Area = length \times width}$

You can find the length and width of the rectangle by using the distance formula or by using other geometric properties of the rectangle.

Q: Can I use the distance formula to find the area of a rectangle?

A: No, the distance formula is used to find the distance between two points, not the area of a rectangle. To find the area of a rectangle, you need to use the formula for the area of a rectangle, which is given by:

${Area = length \times width}$

Q: What if the rectangle is not a perfect rectangle? Can I still compute the area?

A: Yes, you can still compute the area of a rectangle even if it is not a perfect rectangle. You can use the formula for the area of a rectangle, which is given by:

${Area = length \times width}$

You can find the length and width of the rectangle by using the distance formula or by using other geometric properties of the rectangle.

Q: Can I use a calculator to compute the area of a rectangle?

A: Yes, you can use a calculator to compute the area of a rectangle. Simply plug in the values of the length and width into the formula for the area of a rectangle, and the calculator will give you the result.

Q: What if I make a mistake when computing the area of a rectangle? Can I still get the correct answer?

A: Yes, you can still get the correct answer even if you make a mistake when computing the area of a rectangle. Simply recheck your work and make sure that you have used the correct formula and values.

Conclusion

In this article, we have answered some frequently asked questions about computing the area of a rectangle with given vertices. We have covered topics such as the distance formula, finding the length of the sides, and using a calculator to compute the area. We hope that this article has been helpful in answering your questions and providing you with a better understanding of how to compute the area of a rectangle with given vertices.