Plotting Vertices And Finding The Area Of A Rectangle

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In this mathematical exploration, we embark on a journey to understand the properties of a rectangle when given three of its vertices. We will delve into the process of plotting these vertices on a coordinate plane, determining the location of the missing vertex, and calculating the area of the resulting rectangle. This exercise not only reinforces our understanding of geometric concepts but also enhances our problem-solving skills in coordinate geometry.

Plotting the Vertices

Our initial task is to plot the given vertices on a coordinate plane. The vertices provided are (-2, -8), (-2, 2), and (8, 2). To accurately plot these points, we need to understand the Cartesian coordinate system. The Cartesian plane consists of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is represented by an ordered pair (x, y), where x denotes the point's horizontal position and y denotes its vertical position.

  • The first vertex, (-2, -8), indicates that we need to move 2 units to the left along the x-axis and 8 units down along the y-axis. This point lies in the third quadrant of the coordinate plane, where both x and y values are negative.
  • The second vertex, (-2, 2), requires us to move 2 units to the left along the x-axis and 2 units up along the y-axis. This point lies in the second quadrant, where x is negative and y is positive.
  • The third vertex, (8, 2), directs us to move 8 units to the right along the x-axis and 2 units up along the y-axis. This point is located in the first quadrant, where both x and y values are positive.

By plotting these three vertices, we begin to visualize the rectangle taking shape. The next step is to identify the location of the missing fourth vertex, which will complete the rectangular form.

Finding the Last Vertex

To determine the coordinates of the missing fourth vertex, we must utilize our understanding of rectangles and their properties. A rectangle is a quadrilateral with four right angles and opposite sides that are equal in length. This means that the sides of a rectangle are parallel to the coordinate axes. Given the three vertices (-2, -8), (-2, 2), and (8, 2), we can deduce the position of the fourth vertex by considering the properties of a rectangle.

Let's analyze the given vertices:

  • The vertices (-2, -8) and (-2, 2) share the same x-coordinate (-2), which indicates that they lie on the same vertical line. This line segment forms one side of the rectangle.
  • The vertices (-2, 2) and (8, 2) share the same y-coordinate (2), which means they lie on the same horizontal line. This line segment forms another side of the rectangle.

Since opposite sides of a rectangle are parallel and equal in length, we can infer the coordinates of the fourth vertex. The missing vertex must form a side parallel to the side connecting (-2, -8) and (-2, 2), and another side parallel to the side connecting (-2, 2) and (8, 2). To do this, we find a point that has the same x-coordinate as (8, 2) and the same y-coordinate as (-2,-8).

  • The x-coordinate of the fourth vertex must be the same as the x-coordinate of (8, 2), which is 8.
  • The y-coordinate of the fourth vertex must be the same as the y-coordinate of (-2, -8), which is -8.

Therefore, the coordinates of the missing fourth vertex are (8, -8). This point completes the rectangle, ensuring that all four angles are right angles and opposite sides are parallel and equal in length. Now that we have all four vertices, we can proceed to calculate the area of the rectangle. This calculation will require us to determine the lengths of the sides of the rectangle using the coordinates of the vertices.

Calculating the Area of the Rectangle

Now that we have all four vertices of the rectangle: (-2, -8), (-2, 2), (8, 2), and (8, -8), our next objective is to calculate the area of the rectangle. The area of a rectangle is given by the product of its length and width. To find the length and width, we can use the distance formula or simply count the units between the vertices on the coordinate plane.

Let's consider the sides formed by the vertices:

  • Side 1: The side connecting (-2, -8) and (-2, 2). Since these points have the same x-coordinate, the length of this side is the difference in their y-coordinates. Length = |2 - (-8)| = |2 + 8| = 10 units.
  • Side 2: The side connecting (-2, 2) and (8, 2). Since these points have the same y-coordinate, the length of this side is the difference in their x-coordinates. Width = |8 - (-2)| = |8 + 2| = 10 units.

Now that we have the length and width of the rectangle, we can calculate the area. The formula for the area of a rectangle is Area = Length × Width.

  • Area = 10 units × 10 units = 100 square units.

Thus, the area of the rectangle formed by the given vertices is 100 square units. This calculation concludes our geometric exploration, demonstrating how to plot vertices, identify missing vertices, and calculate the area of a rectangle on the coordinate plane. Understanding these concepts is crucial for further studies in geometry and related fields. The process of visualizing shapes on a coordinate plane and applying geometric formulas enhances our spatial reasoning and problem-solving abilities. In summary, this exercise not only provides a solution to the specific problem but also reinforces fundamental mathematical principles.

In conclusion, by plotting the vertices (-2, -8), (-2, 2), and (8, 2), we determined the last vertex of the rectangle to be (8, -8). Furthermore, we calculated the area of the rectangle to be 100 square units. This exercise highlights the importance of understanding coordinate geometry and its applications in solving geometric problems. The ability to plot points, visualize shapes, and apply relevant formulas is a valuable skill in mathematics and various other disciplines.