Quadrilateral ABCD has coordinates A (3, −5), B (5, −2), C (10, −4), D (8, −7). Quadrilateral ABCD is a (4 points)

a
rectangle, because opposite sides are congruent and adjacent sides are perpendicular

b
square, because all four sides are congruent and adjacent sides are perpendicular

c
parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular

d
rhombus, because all four sides are congruent and adjacent sides are not perpendicular

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).