Before we begin, note that we need to use the distanse formula for everything to reach our conclusion. With that being said, here we go:
[tex]\displaystyle \sqrt{[-x_1 + x_2]^2 + [-y_1 + y_2]^2} = d[/tex]
Using the ordered pairs [tex]\displaystyle [3, -3][/tex] and [tex]\displaystyle [4, -2][/tex] for instanse:
[tex]\displaystyle \sqrt{[-3 + 4]^2 + [3 - 2]^2} = \sqrt{1^2 + 1^2} = \sqrt{2}[/tex]
Now, sinse the distanse between [tex]\displaystyle [3, -3][/tex] and [tex]\displaystyle [4, -2][/tex] is [tex]\displaystyle \sqrt{2}[/tex] units, then the distanse between [tex]\displaystyle [1, 1][/tex] and [tex]\displaystyle [0, 0][/tex] ALSO has to be [tex]\displaystyle \sqrt{2}[/tex] units. By definition, lettre c has already been answered for you because sinse it is a rectangle, if the two short sides are congruent, then the two long sides ALSO have to be congruent, but just in case you want to be sure [you do not trust your instincts], just simply re-use the distanse formula.
Using the ordered pairs [tex]\displaystyle [1, 1][/tex] and [tex]\displaystyle [4, -2][/tex] for instanse:
[tex]\displaystyle \sqrt{[-1 + 4]^2 + [-1 - 2]^2} = \sqrt{3^2 + [-3]^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}[/tex]
So there you have it. The length of both of the long sides is [tex]\displaystyle 3\sqrt{2}[/tex] units.
Now that we cleared all of that up, we can now find the perimetre and area of this rectangle:
[tex]\displaystyle 2w + 2l = P[/tex]
[tex]\displaystyle 2[\sqrt{2}] + 2[3\sqrt{2}] = 2\sqrt{2} + 6\sqrt{2} = 8\sqrt{2}[/tex]
The perimetre of this rectangle is [tex]\displaystyle 8\sqrt{2}[/tex] units.
[tex]\displaystyle wl = A[/tex]
[tex]\displaystyle [\sqrt{2}][3\sqrt{2}] = [3][2] = 6[/tex]
The area of this rectangle is [tex]\displaystyle 6[/tex] squared units.
You have now found what you were looking for.
** All rectangles are parallelograms because they both have two pairs of parallel and congruent sides, while vice versa is falce because a parallelogram does not have four congruent right angles, so it is safe to say that this is both a rectangle AND parallelogram, sinse the picture displayed is a rectangle.
I am joyous to assist you at any time. ☺️
