Question

Is the following shape a square? How do you know?

A. No, opposite sides are not parallel.
B. No, the sides are not congruent.
C. Yes, the opposite sides are parallel, and all sides are the same length.
D. No, adjacent sides are not perpendicular.

Answer 1

The answer would be C. Hope this helps.

Answer 2

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=√((-3+1)^2+(0-2)^2)=√(4+4)=√(8)=2\sqrt2,\\\\BC=√((-1+3)^2+(-2-0)^2)=√(4+4)=√(8)=2\sqrt2,\\\\CD=√((1+1)^2+(0+2)^2)=√(4+4)=\sqrt 8=2\sqrt2,\\\\DA=√((-1-1)^2+(2-0)^2)=√(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=(0-2)/(-3+1)=1,\\\\\textup{slope of BC, }n=(-2-0)/(-1+3)=-1,\\\\\textup{slope of CD, }o=(0+2)/(1+1)=1,\\\\\textup{slope of DA, }p=(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.