Question

Quadrilateral WXYZ has vertices W(−1, 5), X(2, 8), Y(4, 4), and Z(1, 1). Using at least one of the properties of parallelograms, show steps or explanation proving that WXYZ is a parallelogram.

Answer 1

To prove that quadrilateral WXYZ is a parallelogram, we can use the property that states that a quadrilateral is a parallelogram if the midpoints of its diagonals are the same.

### Steps to Find the Midpoints of the Diagonals

1. **Find the coordinates of the diagonals**:
The diagonals of quadrilateral WXYZ are WY and XZ.

- W(−1, 5) to Y(4, 4) is the first diagonal (WY).
- X(2, 8) to Z(1, 1) is the second diagonal (XZ).

2. **Calculate the midpoint of WY**:
The midpoint formula is given by:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the segment.

Applying this to WY:
\[
\text{Midpoint of WY} = \left( \frac{-1 + 4}{2}, \frac{5 + 4}{2} \right) = \left( \frac{3}{2}, \frac{9}{2} \right)
\]

3. **Calculate the midpoint of XZ**:
Again, using the midpoint formula:
\[
\text{Midpoint of XZ} = \left( \frac{2 + 1}{2}, \frac{8 + 1}{2} \right) = \left( \frac{3}{2}, \frac{9}{2} \right)
\]

### Conclusion

Both midpoints are the same:
\[
\text{Midpoint of WY} = \left( \frac{3}{2}, \frac{9}{2} \right) \quad \text{and} \quad \text{Midpoint of XZ} = \left( \frac{3}{2}, \frac{9}{2} \right)
\]

Since the midpoints of both diagonals (WY and XZ) are equal, we have shown that the diagonals bisect each other. Therefore, quadrilateral WXYZ is a parallelogram.