Name the polygon that can be graphed in the coordinate plane with vertices at negative two, three; three, three; three, negative one; and negative two, negative one.
Let’s plot these four coordinates that we’re given. So, when we have plotted the coordinates and joined them together, we can see that it does look very much like a rectangle. The definition of a rectangle is that it’s a quadrilateral, or four-sided shape, with four right angles. So, we would need to check the size of each of these angles.
Checking the line at the top of this quadrilateral, we can see that since the 𝑦-value remains at three in both coordinates, then this line will be a horizontal line. The same can be said of our line at the base of the quadrilateral since the 𝑦-value in both coordinates is negative one. This means that this line will be horizontal. If we take a look at the line on the left side of the quadrilateral, here the 𝑥-values remain the same in both coordinates, at negative two. Meaning that this line is vertical. We also have a vertical line on the right side of this quadrilateral since both 𝑥-values in these coordinates are three.
We know that horizontal and vertical lines cross at right angles. So, this quadrilateral has four right angles. And therefore, we could say that this polygon is a rectangle. As an additional point, we can notice in our top line that since it goes from negative two to three on our 𝑥-value, then the length of this will be five units. The base of this shape will also have a length of five units.
For our horizontal sides, the 𝑥-value of the coordinates goes from negative one to three, giving us two lines of length four. This means that our polygon has two pairs of congruent sides. This is another common property of rectangles but wouldn’t be sufficient alone to prove that it is a rectangle. We would need to prove that there are four right angles, as we’ve already demonstrated, to show that it is indeed a rectangle.