To prove that quadrilateral [tex]\( WXYZ \)[/tex] is a square, we need to show that all four sides are of equal length and that both diagonals are of equal length. Here are the coordinates of the vertices: [tex]\( W(-1, 1) \)[/tex], [tex]\( X(3, 4) \)[/tex], [tex]\( Y(6, 0) \)[/tex], and [tex]\( Z(2, -3) \)[/tex].
1. Calculate the lengths of the sides using the distance formula:
- Distance between [tex]\( W \)[/tex] and [tex]\( X \)[/tex]:
[tex]\[
WX = \sqrt{(3 - (-1))^2 + (4 - 1)^2} = \sqrt{(3 + 1)^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5.0
\][/tex]
- Distance between [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
[tex]\[
XY = \sqrt{(6 - 3)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0
\][/tex]
- Distance between [tex]\( Y \)[/tex] and [tex]\( Z \)[/tex]:
[tex]\[
YZ = \sqrt{(2 - 6)^2 + (-3 - 0)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.0
\][/tex]
- Distance between [tex]\( Z \)[/tex] and [tex]\( W \)[/tex]:
[tex]\[
ZW = \sqrt{(2 - (-1))^2 + (-3 - 1)^2} = \sqrt{(2 + 1)^2 + (-3 - 1)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0
\][/tex]
2. Calculate the lengths of the diagonals using the distance formula:
- Distance between [tex]\( W \)[/tex] and [tex]\( Y \)[/tex]:
[tex]\[
WY = \sqrt{(6 - (-1))^2 + (0 - 1)^2} = \sqrt{(6 + 1)^2 + (0 - 1)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} \approx 7.071
\][/tex]
- Distance between [tex]\( X \)[/tex] and [tex]\( Z \)[/tex]:
[tex]\[
XZ = \sqrt{(2 - 3)^2 + (-3 - 4)^2} = \sqrt{(2 - 3)^2 + (-3 - 4)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} \approx 7.071
\][/tex]
3. Verify the conditions for the quadrilateral to be a square:
- Check if all sides are equal:
[tex]\[
WX = XY = YZ = ZW = 5.0
\][/tex]
- Check if both diagonals are equal:
[tex]\[
WY \approx 7.071 \text{ and } XZ \approx 7.071
\][/tex]
Since all four sides are equal and both diagonals are equal, quadrilateral [tex]\( WXYZ \)[/tex] satisfies the conditions for being a square. Therefore, [tex]\( WXYZ \)[/tex] is a square.
So, we can select the correct answers:
- [tex]\( WX = XY = YZ = ZW = 5.0 \)[/tex]
- [tex]\( WY \approx XZ \approx 7.071 \)[/tex]
- [tex]\( WXYZ \)[/tex] is a square.
