The Vertices Of A Rectangle Are At $(-4,2), (3,2), (3,-2$\], And $(-4,-2$\]. What Is The Length Of The Longer Side Of The Rectangle?Enter Your Answer In The Box.$\square$ Units

Introduction

In geometry, a rectangle is a type of quadrilateral with four right angles and opposite sides of equal length. The vertices of a rectangle are the points where the sides meet, and they can be used to calculate various properties of the rectangle, such as its length and width. In this article, we will explore how to calculate the length of the longer side of a rectangle given its vertices.

Understanding the Problem

The problem presents a rectangle with vertices at $(-4,2), (3,2), (3,-2), and $(-4,-2)$. To calculate the length of the longer side, we need to find the distance between two opposite vertices. The distance formula is given by:

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

where $d$ is the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$.

Calculating the Distance Between Two Opposite Vertices

Let's calculate the distance between the vertices $(-4,2)$ and $(3,2)$. We can use the distance formula:

$$d = \sqrt{(3 - (-4))^2 + (2 - 2)^2}$$

$$d = \sqrt{(7)^2 + (0)^2}$$

$$d = \sqrt{49}$$

$$d = 7$$

This means that the distance between the vertices $(-4,2)$ and $(3,2)$ is 7 units.

Calculating the Distance Between the Other Two Opposite Vertices

Now, let's calculate the distance between the vertices $(3,2)$ and $(3,-2)$. We can use the distance formula:

$$d = \sqrt{(3 - 3)^2 + (-2 - 2)^2}$$

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

$$d = \sqrt{16}$$

$$d = 4$$

This means that the distance between the vertices $(3,2)$ and $(3,-2)$ is 4 units.

Comparing the Distances

We have calculated the distances between two pairs of opposite vertices. The first pair has a distance of 7 units, and the second pair has a distance of 4 units. Since the longer side of the rectangle is the one with the greater distance, we can conclude that the longer side of the rectangle has a length of 7 units.

Conclusion

In this article, we have explored how to calculate the length of the longer side of a rectangle given its vertices. We used the distance formula to calculate the distances between two pairs of opposite vertices and compared them to determine the length of the longer side. The result is that the longer side of the rectangle has a length of 7 units.

Final Answer

The final answer is: $\boxed{7}$

Introduction

In our previous article, we explored how to calculate the length of the longer side of a rectangle given its vertices. We used the distance formula to calculate the distances between two pairs of opposite vertices and compared them to determine the length of the longer side. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the distance formula?

A: The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate plane. It is given by:

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

where $d$ is the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$.

Q: How do I calculate the distance between two points in a coordinate plane?

A: To calculate the distance between two points in a coordinate plane, you can use the distance formula. Simply substitute the coordinates of the two points into the formula and simplify.

Q: What is the difference between the distance formula and the Pythagorean theorem?

A: The distance formula and the Pythagorean theorem are both used to calculate the distance between two points in a coordinate plane. However, the Pythagorean theorem is used to calculate the distance between two points in a right-angled triangle, whereas the distance formula is used to calculate the distance between two points in a coordinate plane.

Q: How do I determine the length of the longer side of a rectangle?

A: To determine the length of the longer side of a rectangle, you need to calculate the distances between two pairs of opposite vertices. The longer side is the one with the greater distance.

Q: What if the rectangle is not a standard rectangle with equal sides?

A: If the rectangle is not a standard rectangle with equal sides, you can still use the distance formula to calculate the distances between two pairs of opposite vertices. The longer side is the one with the greater distance.

Q: Can I use the distance formula to calculate the distance between two points in 3D space?

A: Yes, you can use the distance formula to calculate the distance between two points in 3D space. However, you need to use the 3D distance formula, which is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

Q: What if I have a rectangle with vertices at $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, and $(x_4, y_4)$?

A: To calculate the length of the longer side of a rectangle with vertices at $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, and $(x_4, y_4)$, you need to calculate the distances between two pairs of opposite vertices. The longer side is the one with the greater distance.

Conclusion

In this article, we have answered some frequently asked questions related to calculating the length of the longer side of a rectangle given its vertices. We have used the distance formula to calculate the distances between two pairs of opposite vertices and compared them to determine the length of the longer side. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is: $\boxed{7}$