REGULARITY  Determine whether quadrilateral WXYZ is a rhombus, a rectangle, a square, a parallelogram, or none. Select all that apply. Explain. W(–3𝑊(–3 , 2)2) , X(2𝑋(2 , 5)5) , Y(4𝑌(4 , 2)2) , Z(–1𝑍(–1 , –2)–2)

To determine the type of quadrilateral WXYZ, we need to calculate the distances between the points.

The distance between two points (x1, y1) and (x2, y2) can be calculated using the formula: √[(x2 - x1)² + (y2 - y1)²]

1. Calculate the distance between W and X: d(W, X) = √[(2 - (-3))² + (5 - 2)²] = √[25 + 9] = √34

2. Calculate the distance between X and Y: d(X, Y) = √[(4 - 2)² + (2 - 5)²] = √[4 + 9] = √13

3. Calculate the distance between Y and Z: d(Y, Z) = √[(-1 - 4)² + (-2 - 2)²] = √[25 + 16] = √41

4. Calculate the distance between Z and W: d(Z, W) = √[(-3 - (-1))² + (2 - (-2))²] = √[4 + 16] = √20

5. Calculate the distance between W and Y (diagonal): d(W, Y) = √[(4 - (-3))² + (2 - 2)²] = √[49 + 0] = √49 = 7

6. Calculate the distance between X and Z (diagonal): d(X, Z) = √[(-1 - 2)² + (-2 - 5)²] = √[9 + 49] = √58

Since the distances between W and X, X and Y, Y and Z, and Z and W are not equal, WXYZ is not a rhombus or a square.

Since the diagonals WY and XZ are not equal, WXYZ is not a rectangle.

Since the opposite sides WX and YZ, and XY and ZW are not equal, WXYZ is not a parallelogram.

Therefore, quadrilateral WXYZ is none of the above (not a rhombus, rectangle, square, or parallelogram).