Answer:
- point: (-5, 2)
- length: 8 units
- width: 5 units
- area: 40 square units
Explanation:
Given 3 points, you want the fourth point that defines a rectangle. Then you want the length, width, and area of the rectangle. {(3, 2), (3, -3), (-5, -3)}
Rectangle
A rectangle is a quadrilateral (4-sided polygon) with opposite sides the same length and parallel, and all corners 90°.
Points
We can name the given points A(3, 2), B(3, -3), and C(-5, -3).
When you plot the given points, you can imagine they form a right triangle with the right angle at B. The missing sides of the rectangle are ...
A line parallel to BC through A, and a line parallel to AB through C.
The equation of the first line is y = 2, and the equation of the second line is x = -5. These two lines meet at the point (x, y) = (-5, 2)
The missing point is (-5, 2).
Length
We generally think of "length" as being the longer dimension of the figure. Here, the points B and C are farther apart than the points A and B. The distance between those points is the difference of their x-coordinates:
length = 3 -(-5) = 8
The length of the rectangle is 8 units.
Width
As with length, we can determine the distance between points A and B as the difference of their y-coordinates:
width = 2 -(-3) = 5
The width of the rectangle is 5 units.
Area
The area of a rectangle is the product of its length and width:
A = LW
A = (8)(5) = 40 . . . . square units
The area of the rectangle is 40 square units.
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Additional comment
When the sides of the rectangle are on grid lines, as here, finding the missing point and the various lengths is pretty easy.
You will notice that there are two different x-coordinates and two different y-coordinates among the four points. The four points are those coordinates in every combination. One way to find the missing point is to find the missing combination. The side lengths are the difference between x-coordinates and the difference between y-coordinates. (This works for rectangles aligned to the grid.)
When the sides are skew with respect to the grid lines, a different strategy for locating the missing point and for finding side lengths is required. That strategy can make use of the fact that the diagonals cross at their midpoints, and the length is given by the distance formula.
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