Question

select the correct answer from each drop - down menu.
wxyz is a quadrilateral graphed in the coordinate plane with vertices w(0,5), x(-3,2), y(0,-4), and z(3,2). what are the lengths of the sides of the quadrilateral, and what is the correct name for the figure?
the length of (overline{wx}) is about (\boldsymbol{downarrow}). the length of (overline{xy}) is about (\boldsymbol{downarrow}).
the length of (overline{yz}) is about (\boldsymbol{downarrow}). the length of (overline{zw}) is about (\boldsymbol{downarrow}).
the best name for this quadrilateral is (\boldsymbol{downarrow}).
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Explanation:

Step1: Recall Distance Formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate \(WX\) (W(0,5), X(-3,2))

\[ \begin{align*} WX&=\sqrt{(-3 - 0)^2+(2 - 5)^2}\\ &=\sqrt{(-3)^2+(-3)^2}\\ &=\sqrt{9 + 9}\\ &=\sqrt{18}\\ &\approx 4.24 \end{align*} \]

Step3: Calculate \(XY\) (X(-3,2), Y(0,-4))

\[ \begin{align*} XY&=\sqrt{(0 - (-3))^2+(-4 - 2)^2}\\ &=\sqrt{(3)^2+(-6)^2}\\ &=\sqrt{9 + 36}\\ &=\sqrt{45}\\ &\approx 6.71 \end{align*} \]

Step4: Calculate \(YZ\) (Y(0,-4), Z(3,2))

\[ \begin{align*} YZ&=\sqrt{(3 - 0)^2+(2 - (-4))^2}\\ &=\sqrt{(3)^2+(6)^2}\\ &=\sqrt{9 + 36}\\ &=\sqrt{45}\\ &\approx 6.71 \end{align*} \]

Step5: Calculate \(ZW\) (Z(3,2), W(0,5))

\[ \begin{align*} ZW&=\sqrt{(0 - 3)^2+(5 - 2)^2}\\ &=\sqrt{(-3)^2+(3)^2}\\ &=\sqrt{9 + 9}\\ &=\sqrt{18}\\ &\approx 4.24 \end{align*} \]

Step6: Analyze Side Lengths

We see \(WX = ZW\approx 4.24\) and \(XY = YZ\approx 6.71\). Also, check slopes for perpendicularity (optional, but for rhombus/parallelogram). Slope of \(WX\): \(\frac{2 - 5}{-3 - 0}=1\), slope of \(XY\): \(\frac{-4 - 2}{0 - (-3)}=-2\) (Wait, no, earlier calculation: Wait, \(WX\) slope: \(\frac{2 - 5}{-3 - 0}=\frac{-3}{-3}=1\), \(XY\) slope: \(\frac{-4 - 2}{0 - (-3)}=\frac{-6}{3}=-2\)? Wait no, mistake. Wait \(X(-3,2)\) to \(Y(0,-4)\): \(y\) change: \(-4 - 2=-6\), \(x\) change: \(0 - (-3)=3\), slope \(-2\). \(Y(0,-4)\) to \(Z(3,2)\): \(y\) change \(6\), \(x\) change \(3\), slope \(2\). \(Z(3,2)\) to \(W(0,5)\): \(y\) change \(3\), \(x\) change \(-3\), slope \(-1\). Wait, but side lengths: \(WX\approx 4.24\), \(XY\approx 6.71\), \(YZ\approx 6.71\), \(ZW\approx 4.24\). So opposite sides equal. So it's a parallelogram. Also, check if adjacent sides are equal? No, \(4.24 eq 6.71\). Wait, but \(WX = ZW\) and \(XY = YZ\)? Wait no, \(WX\) and \(ZW\) are adjacent? Wait vertices order: W(0,5), X(-3,2), Y(0,-4), Z(3,2). So sides: WX, XY, YZ, ZW. So WX: W to X, XY: X to Y, YZ: Y to Z, ZW: Z to W. So WX and YZ: WX≈4.24, YZ≈6.71? Wait no, earlier calculation: YZ: Y(0,-4) to Z(3,2): \(\sqrt{(3 - 0)^2+(2 - (-4))^2}=\sqrt{9 + 36}=\sqrt{45}\approx 6.71\). WX: W(0,5) to X(-3,2): \(\sqrt{(-3 - 0)^2+(2 - 5)^2}=\sqrt{9 + 9}=\sqrt{18}\approx 4.24\). XY: X(-3,2) to Y(0,-4): \(\sqrt{(0 - (-3))^2+(-4 - 2)^2}=\sqrt{9 + 36}=\sqrt{45}\approx 6.71\). ZW: Z(3,2) to W(0,5): \(\sqrt{(0 - 3)^2+(5 - 2)^2}=\sqrt{9 + 9}=\sqrt{18}\approx 4.24\). Ah! So WX = ZW (\(\approx 4.24\)) and XY = YZ (\(\approx 6.71\))? Wait no, WX is W to X, ZW is Z to W. Wait, WX: W(0,5)-X(-3,2), ZW: Z(3,2)-W(0,5). So WX and ZW: WX: \(\sqrt{(-3)^2+(-3)^2}=\sqrt{18}\), ZW: \(\sqrt{(-3)^2+3^2}=\sqrt{18}\). So WX = ZW. XY: X(-3,2)-Y(0,-4): \(\sqrt{3^2+(-6)^2}=\sqrt{45}\), YZ: Y(0,-4)-Z(3,2): \(\sqrt{3^2+6^2}=\sqrt{45}\). So XY = YZ. So two pairs of adjacent sides equal? Wait, WX and ZW are adjacent? No, WX is from W to X, XY is from X to Y, YZ from Y to Z, ZW from Z to W. So WX and XY are adjacent, XY and YZ, YZ and ZW, ZW and WX. So WX = ZW (\(\sqrt{18}\)) and XY = YZ (\(\sqrt{45}\)). So this is a kite? Wait no, kite has two distinct pairs of adjacent sides equal. Wait WX = ZW, XY = YZ. So yes, two pairs of adjacent sides equal. Wait but also, check diagonals. Diagonal WY: W(0,5) to Y(0,-4): length is \(|5 - (-4)| = 9\) (vertical line). Diagonal XZ: X(-3,2) to Z(3,2): length is \(|3 - (-3)| = 6\) (horizontal line). So diagonals are perpendicular? Slope of WY: vertical line (undefined), slope of XZ: horizontal line (0), so they are perpendicular. So it's a kite? Wait but also, if diagonals are perpendicular and one diagonal is bisected? Wait WY is from (0,5) to (0,-4), midpoint at (0, 0.5). XZ is from (-3,2) to (3,2), midpoint at (0,2). Not the same midpoint. Wait, but in a kite, one diagonal is bisected by the other. Wait maybe I made a mistake in side lengths. Wait let's recalculate WX: W(0,5), X(-3,2): \(\sqrt{(-3 - 0)^2 + (2 - 5)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24\). XY: X(-3,2), Y(0,-4): \(\sqrt{(0 - (-3))^2 + (-4 - 2)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71\). YZ: Y(0,-4), Z(3,2): \(\sqrt{(3 - 0)^2 + (2 - (-4))^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71\). ZW: Z(3,2), W(0,5): \(\sqrt{(0 - 3)^2 + (5 - 2)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24\). So WX = ZW, XY = YZ. So two pairs of adjacent sides equal. So it's a kite. Also, diagonals are perpendicular (WY is vertical, XZ is horizontal). So the quadrilateral is a kite. Wait but also, if we check the lengths again: WX ≈ 4.24, XY ≈ 6.71, YZ ≈ 6.71, ZW ≈ 4.24. So the lengths: WX ≈ 4.24, XY ≈ 6.71, YZ ≈ 6.71, ZW ≈ 4.24. And the best name is a kite (or maybe a rhombus? No, sides are not all equal). Wait, no, WX and ZW are equal, XY and YZ are equal, but WX ≠ XY. So it's a kite.

Answer:

• Length of \( \overline{WX} \approx 4.24 \)
• Length of \( \overline{XY} \approx 6.71 \)
• Length of \( \overline{YZ} \approx 6.71 \)
• Length of \( \overline{ZW} \approx 4.24 \)
• Best name: Kite (or check options, but based on calculation, the side lengths are \(WX\approx 4.24\), \(XY\approx 6.71\), \(YZ\approx 6.71\), \(ZW\approx 4.24\), and the quadrilateral is a kite (or possibly a rhombus? No, sides not all equal; or a parallelogram? No, opposite sides equal? Wait WX and YZ: WX≈4.24, YZ≈6.71, not equal. XY and ZW: XY≈6.71, ZW≈4.24, not equal. Wait, wait! I made a mistake earlier. WX is W to X, YZ is Y to Z. WX: \(\sqrt{18}\), YZ: \(\sqrt{45}\) – not equal. XY: X to Y, ZW: Z to W. XY: \(\sqrt{45}\), ZW: \(\sqrt{18}\) – not equal. Wait, so my earlier mistake: WX and ZW are adjacent, XY and YZ are adjacent. So WX = ZW (\(\sqrt{18}\)), XY = YZ (\(\sqrt{45}\)). So adjacent sides equal, opposite sides not equal. So it's a kite. And diagonals are perpendicular (WY is vertical, XZ is horizontal). So the lengths: \(WX\approx 4.24\), \(XY\approx 6.71\), \(YZ\approx 6.71\), \(ZW\approx 4.24\), and the quadrilateral is a kite (or if options have "kite" or "rhombus" or "parallelogram", but based on side lengths, two pairs of adjacent sides equal, so kite).