To solve the problem, let's determine the lengths of each side of the quadrilateral [tex]\(WXYZ\)[/tex], and then identify the type of figure it forms.
### Step 1: Calculating the Lengths of Each Side
The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
#### Length of [tex]\(\overline{WX}\)[/tex]
[tex]\[
W(0,5), X(-3,2)
\][/tex]
[tex]\[
\text{Length of } \overline{WX} = \sqrt{(0 - (-3))^2 + (5 - 2)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24
\][/tex]
#### Length of [tex]\(\overline{XY}\)[/tex]
[tex]\[
X(-3,2), Y(0,-4)
\][/tex]
[tex]\[
\text{Length of } \overline{XY} = \sqrt{(0 - (-3))^2 + (-4 - 2)^2} = \sqrt{3^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71
\][/tex]
#### Length of [tex]\(\overline{YZ}\)[/tex]
[tex]\[
Y(0,-4), Z(3,2)
\][/tex]
[tex]\[
\text{Length of } \overline{YZ} = \sqrt{(3 - 0)^2 + (2 - (-4))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71
\][/tex]
#### Length of [tex]\(\overline{ZW}\)[/tex]
[tex]\[
Z(3,2), W(0,5)
\][/tex]
[tex]\[
\text{Length of } \overline{ZW} = \sqrt{(0 - 3)^2 + (5 - 2)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24
\][/tex]
### Step 2: Determining the Best Name for the Quadrilateral
To classify the quadrilateral, we observe the lengths of the sides and the diagonals.
Both pairs of opposite sides are equal ([tex]\(\overline{WX} = \overline{ZW}\)[/tex] and [tex]\(\overline{XY} = \overline{YZ}\)[/tex]). In a square or rectangle, the diagonals are equal. Let's check the diagonals.
[tex]\[
\text{Diagonal } \overline{WY} = \sqrt{(0 - 0)^2 + (5 - (-4))^2} = \sqrt{0 + 81} = \sqrt{81} = 9
\][/tex]
[tex]\[
\text{Diagonal } \overline{XZ} = \sqrt{(3 - (-3))^2 + (2 - 2)^2} = \sqrt{6^2 + 0} = \sqrt{36} = 6
\][/tex]
Since the diagonals are not equal ([tex]\(\overline{WY} \neq \overline{XZ}\)[/tex]), this quadrilateral is not a square or rectangle. Therefore, the best name for this quadrilateral is simply a quadrilateral.
### Final Answer
The length of [tex]\(\overline{WX}\)[/tex] is about 4.24.
The length of [tex]\(\overline{XY}\)[/tex] is about 6.71.
The length of [tex]\(\overline{YZ}\)[/tex] is about 6.71.
The length of [tex]\(\overline{ZW}\)[/tex] is about 4.24.
The best name for this quadrilateral is quadrilateral.
Using the drop-down menus:
- The length of [tex]\(\overline{WX}\)[/tex] is about 4.24.
- The length of [tex]\(\overline{XY}\)[/tex] is about 6.71.
- The length of [tex]\(\overline{YZ}\)[/tex] is about 6.71.
- The length of [tex]\(\overline{ZW}\)[/tex] is about 4.24.
- The best name for this quadrilateral is quadrilateral.
