Find the area and perimeter of a rectangle for which (– 2, 2), (7,2), and (7, – 3) are three of the vertices. What is the 4th vertex?
You could sketch the given points, but ...
just looking at the values, we can see the base runs from -2 to 7, so the base is 9
and the height goes from -3 to 2, so the height is 5
area = 5(9) = 45
perimeter = 2(5) + 2(9) = 28
missing point must be (-2, -3)
To find the area and perimeter of a rectangle, we need to first determine the length and width of the rectangle. Given three vertices, we can determine the length and width by finding the distances between these points.
Let's find the length and width first:
Length:
The length of a rectangle is the distance between two vertices along the horizontal axis. Given the points (–2, 2) and (7, 2), we can calculate the length as follows:
Length = |x2 - x1| = |7 - (-2)| = |7 + 2| = 9
Width:
The width of a rectangle is the distance between two vertices along the vertical axis. Given the points (7, 2) and (7, -3), we can calculate the width as follows:
Width = |y2 - y1| = |-3 - 2| = |-3 + 2| = 5
Now that we have the length and width, we can find the area and perimeter.
Area of a rectangle:
The area of a rectangle is calculated by multiplying the length by the width:
Area = Length * Width = 9 * 5 = 45 square units
Perimeter of a rectangle:
The perimeter of a rectangle is calculated by adding the lengths of all its sides. Since a rectangle has two pairs of equal sides, we can use the formula:
Perimeter = 2 * (Length + Width) = 2 * (9 + 5) = 2 * 14 = 28 units
Now, to find the fourth vertex, we know that opposite vertices of a rectangle are symmetric about the center. In this case, the center is the midpoint of the diagonal connecting the given vertices. The midpoint can be found by averaging the x-coordinates and y-coordinates of the given vertices.
Average of x-coordinates = (-2 + 7) / 2 = 5 / 2 = 2.5
Average of y-coordinates = (2 - 3) / 2 = -1 / 2 = -0.5
So, the coordinates of the fourth vertex would be (2.5, -0.5).