To solve this problem, we need to calculate the distances between each pair of adjacent vertices and then determine the proper geometric name for the quadrilateral based on those side lengths.
1. Calculate the Lengths of the Sides:
- For side [tex]\( \overline{WX} \)[/tex]:
The coordinates of [tex]\( W \)[/tex] are [tex]\( (0, 5) \)[/tex] and [tex]\( X \)[/tex] are [tex]\( (-3, 2) \)[/tex].
Using the distance formula:
[tex]\[
\sqrt{(W_x - X_x)^2 + (W_y - X_y)^2}
= \sqrt{(0 - (-3))^2 + (5 - 2)^2}
= \sqrt{3^2 + 3^2}
= \sqrt{9 + 9}
= \sqrt{18}
= 4.242640687119285
\][/tex]
- For side [tex]\( \overline{XY} \)[/tex]:
The coordinates of [tex]\( X \)[/tex] are [tex]\( (-3, 2) \)[/tex] and [tex]\( Y \)[/tex] are [tex]\( (0, -4) \)[/tex].
Using the distance formula:
[tex]\[
\sqrt{(X_x - Y_x)^2 + (X_y - Y_y)^2}
= \sqrt{(-3 - 0)^2 + (2 - (-4))^2}
= \sqrt{(-3)^2 + (6)^2}
= \sqrt{9 + 36}
= \sqrt{45}
= 6.708203932499369
\][/tex]
- For side [tex]\( \overline{YZ} \)[/tex]:
The coordinates of [tex]\( Y \)[/tex] are [tex]\( (0, -4) \)[/tex] and [tex]\( Z \)[/tex] are [tex]\( (3, 2) \)[/tex].
Using the distance formula:
[tex]\[
\sqrt{(Y_x - Z_x)^2 + (Y_y - Z_y)^2}
= \sqrt{(0 - 3)^2 + (-4 - 2)^2}
= \sqrt{(-3)^2 + (-6)^2}
= \sqrt{9 + 36}
= \sqrt{45}
= 6.708203932499369
\][/tex]
- For side [tex]\( \overline{ZW} \)[/tex]:
The coordinates of [tex]\( Z \)[/tex] are [tex]\( (3, 2) \)[/tex] and [tex]\( W \)[/tex] are [tex]\( (0, 5) \)[/tex].
Using the distance formula:
[tex]\[
\sqrt{(Z_x - W_x)^2 + (Z_y - W_y)^2}
= \sqrt{(3 - 0)^2 + (2 - 5)^2}
= \sqrt{3^2 + (-3)^2}
= \sqrt{9 + 9}
= \sqrt{18}
= 4.242640687119285
\][/tex]
2. Determine the Name of the Quadrilateral:
We have calculated the side lengths to be approximately:
[tex]\[
\overline{WX} \approx 4.242640687119285, \quad \overline{XY} \approx 6.708203932499369, \quad \overline{YZ} \approx 6.708203932499369, \quad \overline{ZW} \approx 4.242640687119285
\][/tex]
Observing that [tex]\( \overline{WX} \approx \overline{ZW} \)[/tex] and [tex]\( \overline{XY} \approx \overline{YZ} \)[/tex], but not all four sides are equal, we turn it into a trapezoid.
Thus, the best name for this quadrilateral is trapezoid.
The completed selections for the drop-down menus are:
- The length of [tex]\( \overline{WX} \)[/tex] is about [tex]\( 4.24 \)[/tex]
- The length of [tex]\( \overline{XY} \)[/tex] is about [tex]\( 6.71 \)[/tex]
- The length of [tex]\( \overline{YZ} \)[/tex] is about [tex]\( 6.71 \)[/tex]
- The length of [tex]\( \overline{ZW} \)[/tex] is about [tex]\( 4.24 \)[/tex]
- The best name for this quadrilateral is trapezoid
