A Rectangle In The Coordinate Plane Has Vertices At \[$(5,8), (2,4),\$\] And \[$(5,4)\$\]. What Are The Coordinates Of The Fourth Vertex?A. \[$(8,5)\$\] B. \[$(8,8)\$\] C. \[$(2,8)\$\] D. \[$(4,5)\$\]

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Understanding the Problem

To find the coordinates of the fourth vertex of a rectangle in the coordinate plane, we need to understand the properties of a rectangle and how it is positioned in the coordinate system. A rectangle is a quadrilateral with four right angles and opposite sides of equal length. In this problem, we are given the coordinates of three vertices of the rectangle: (5,8), (2,4), and (5,4). We need to find the coordinates of the fourth vertex.

Properties of a Rectangle

A rectangle has several key properties that we can use to find the fourth vertex. These properties include:

  • Opposite sides are equal in length and parallel to each other.
  • Adjacent sides are perpendicular to each other.
  • The diagonals of a rectangle bisect each other and are equal in length.

Analyzing the Given Vertices

Let's analyze the given vertices and their positions in the coordinate plane. We have the following vertices:

  • (5,8)
  • (2,4)
  • (5,4)

Finding the Fourth Vertex

To find the fourth vertex, we need to use the properties of a rectangle and the given vertices. We can start by finding the length and width of the rectangle. The length of the rectangle is the distance between the x-coordinates of the given vertices, which is 5 - 2 = 3. The width of the rectangle is the distance between the y-coordinates of the given vertices, which is 8 - 4 = 4.

Using the Properties of a Rectangle

Now that we have the length and width of the rectangle, we can use the properties of a rectangle to find the fourth vertex. Since opposite sides are equal in length and parallel to each other, the fourth vertex must be located at a distance of 3 units from the x-coordinate of the given vertices and 4 units from the y-coordinate of the given vertices.

Possible Locations of the Fourth Vertex

Based on the properties of a rectangle, the fourth vertex can be located in one of the following positions:

  • (2,8)
  • (5,8)
  • (8,4)
  • (5,4)

Eliminating Incorrect Options

We can eliminate some of the options by analyzing the given vertices and their positions in the coordinate plane. For example, if the fourth vertex is located at (5,8), it would not be a rectangle because the opposite sides would not be equal in length. Similarly, if the fourth vertex is located at (8,4), it would not be a rectangle because the opposite sides would not be parallel to each other.

Conclusion

Based on the properties of a rectangle and the given vertices, the fourth vertex must be located at a distance of 3 units from the x-coordinate of the given vertices and 4 units from the y-coordinate of the given vertices. The only option that satisfies this condition is (2,8).

The final answer is C. {(2,8)$}$.

Understanding the Problem

In the previous article, we discussed how to find the coordinates of the fourth vertex of a rectangle in the coordinate plane. We analyzed the properties of a rectangle and used the given vertices to determine the location of the fourth vertex. In this article, we will answer some common questions related to finding the fourth vertex of a rectangle in the coordinate plane.

Q&A

Q: What are the properties of a rectangle that we can use to find the fourth vertex?

A: A rectangle has several key properties that we can use to find the fourth vertex. These properties include:

  • Opposite sides are equal in length and parallel to each other.
  • Adjacent sides are perpendicular to each other.
  • The diagonals of a rectangle bisect each other and are equal in length.

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

A: To find the length and width of the rectangle, we need to find the distance between the x-coordinates and y-coordinates of the given vertices. The length of the rectangle is the distance between the x-coordinates, which is 5 - 2 = 3. The width of the rectangle is the distance between the y-coordinates, which is 8 - 4 = 4.

Q: How do we use the properties of a rectangle to find the fourth vertex?

A: We can use the properties of a rectangle to find the fourth vertex by analyzing the given vertices and their positions in the coordinate plane. Since opposite sides are equal in length and parallel to each other, the fourth vertex must be located at a distance of 3 units from the x-coordinate of the given vertices and 4 units from the y-coordinate of the given vertices.

Q: What are the possible locations of the fourth vertex?

A: Based on the properties of a rectangle, the fourth vertex can be located in one of the following positions:

  • (2,8)
  • (5,8)
  • (8,4)
  • (5,4)

Q: How do we eliminate incorrect options?

A: We can eliminate some of the options by analyzing the given vertices and their positions in the coordinate plane. For example, if the fourth vertex is located at (5,8), it would not be a rectangle because the opposite sides would not be equal in length. Similarly, if the fourth vertex is located at (8,4), it would not be a rectangle because the opposite sides would not be parallel to each other.

Q: What is the final answer?

A: Based on the properties of a rectangle and the given vertices, the fourth vertex must be located at a distance of 3 units from the x-coordinate of the given vertices and 4 units from the y-coordinate of the given vertices. The only option that satisfies this condition is (2,8).

Conclusion

Finding the fourth vertex of a rectangle in the coordinate plane requires a thorough understanding of the properties of a rectangle and how to analyze the given vertices. By using the properties of a rectangle and eliminating incorrect options, we can determine the location of the fourth vertex. In this article, we answered some common questions related to finding the fourth vertex of a rectangle in the coordinate plane.

Additional Resources

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Tags

  • Rectangle
  • Coordinate Plane
  • Geometry
  • Properties of a Rectangle
  • Finding the Fourth Vertex
  • Analyzing the Given Vertices
  • Eliminating Incorrect Options
  • Final Answer