Answer: The correct option is (C). Yes, the opposite sides are parallel, and all sides are the same length.
Step-by-step explanation: We are given to check whether the shape in the figure is a square or not.
From the figure, we note that the co-ordinates of the vertices of shape ABCD are A(-1, 2), B(-3, 0), C(-1, -2) and D(1, 0).
The lengths of the sides of ABCD are calculated by distance formula as follows:
\(AB=\sqrt{(-3+1)^2+(0-2)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt2,\\\\BC=\sqrt{(-1+3)^2+(-2-0)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt2,\\\\CD=\sqrt{(1+1)^2+(0+2)^2}=\sqrt{4+4}=\sqrt 8=2\sqrt2,\\\\DA=\sqrt{(-1-1)^2+(2-0)^2}=\sqrt{4+4}=\sqrt8=2\sqrt2.\)
So, the lengths of all the sides are equal.
We know that the slopes of two parallel lines are equal and slopes of two perpendicular lines have product - 1.
Now, the slopes of the sides of ABCD are given by
\(\textup{slope of AB, }m=\dfrac{0-2}{-3+1}=1,\\\\\textup{slope of BC, }n=\dfrac{-2-0}{-1+3}=-1,\\\\\textup{slope of CD, }o=\dfrac{0+2}{1+1}=1,\\\\\textup{slope of DA, }p=\dfrac{2-0}{-1+-}=-1.\)
Therefore, we have
m = o, n = p, m × n = n × o = o × p = p × m = -1,
which implies that the opposite sides are parallel and adjacent sides are perpendicular.
So, ABCD is a square.
Thus, (C) is the correct option.
