Do points A(2,−1), B(3,4), C(−2,3) and D(−3,2) represent a rhombus?

To determine if the given points form a rhombus, we need to check two conditions:

1. All sides are of equal length.
2. Diagonals are perpendicular bisectors of each other.

Step 1: Calculate the lengths of all sides:

AB = sqrt[(3-2)² + (4-(-1))²] = sqrt[1 + 25] = sqrt[26] BC = sqrt[(-2-3)² + (3-4)²] = sqrt[25 + 1] = sqrt[26] CD = sqrt[(-3-(-2))² + (2-3)²] = sqrt[1 + 1] = sqrt[2] DA = sqrt[(2-(-3))² + ((-1)-2)²] = sqrt[25 + 9] = sqrt[34]

Since AB = BC but not equal to CD = DA, the figure is not a rhombus based on the first condition.

Step 2: Even though we've already determined that the figure is not a rhombus, let's check the second condition for the sake of completeness.

Diagonal AC = sqrt[(2-(-2))² + ((-1)-3)²] = sqrt[16 + 16] = sqrt[32] Diagonal BD = sqrt[(3-(-3))² + (4-2)²] = sqrt[36 + 4] = sqrt[40]

The diagonals are not equal, so they can't be perpendicular bisectors of each other.

Therefore, points A(2,−1), B(3,4), C(−2,3) and D(−3,2) do not represent a rhombus.