9514 1404 393
Answer:
c parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular
Step-by-step explanation:
All of the answer choices are some sort of parallelogram. We know the diagonals of a parallelogram bisect each other. Their lengths are the same when the figure is a rectangle. They are perpendicular when the figure is a rhombus. (The figure is a square if it is both a rectangle and a rhombus.)
We can check the lengths and angles of the diagonals by looking at their direction vectors (the difference between the end points).
AC = C - A = (10, -4) -(3, -5) = (10-3, -4-(-5)) = (7, 1)
BD = D - B = (8, -7) -(5, -2) = (8 -5, -7-(-2)) = (3, -5)
We can see that the sum of squares of these values is different, so the diagonals are different lengths. (7²+1² ≠ 3²+5²) The slopes (ratio of Δy to Δx) do not have a product of -1, so the diagonals are not perpendicular. (1/7×-5/3 ≠ -1)
Hence, the parallelogram is not a rectangle or rhombus (thus, not a square).
The figure is a parallelogram.
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It seems you are expected to compute lengths and slopes of all of the sides. A spreadsheet can help, but it is still a fair amount of work. It is much simpler to compute the midpoints, lengths, and slopes of the diagonals. For a quadrilateral, their relationship tells a lot about the shape. As we have seen here, it isn't always necessary to carry the computation to a final answer, It is sufficient to apply number sense to realize when values do or don't have the right relationships.
Of course, a graph can tell you a lot just by looking (and maybe counting grid squares).
