Question

which statement proves that quadrilateral hijk is a kite?

\\(\overline{hi} \perp \overline{ij}\\), and \\(m\angle h = m\angle j\\).

\\(ih = ij = 3\\) and \\(jk = hk = \sqrt{29}\\), and \\(ih \
eq jk\\) and \\(ij \
eq hk\\).

\\(\overline{ik}\\) intersects \\(\overline{hj}\\) at the midpoint of \\(\overline{hj}\\) at \\((-1.5, 2.5)\\).

the slope of \\(\overline{hk} = -\frac{2}{5}\\) and the slope of \\(\overline{jk} = -\frac{5}{2}\\).

Explanation:

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<reasoning>

Identify the definition of a kite

Using the Kite Properties knowledge point
A kite is defined as a quadrilateral with two distinct pairs of adjacent congruent sides. For quadrilateral \(HIJK\), this means we must show that one pair of adjacent sides is congruent (e.g., \(IH = IJ\)) and the other pair of adjacent sides is congruent (e.g., \(JK = HK\)), while ensuring that opposite sides are not congruent (\(IH
eq JK\) and \(IJ
eq HK\)) to distinguish it from a rhombus.

Find the coordinates of the vertices

We read the coordinates of the vertices from the given coordinate plane:

• \(H = (-3, 1)\)
• \(I = (-3, 4)\)
• \(J = (0, 4)\)
• \(K = (2, -1)\)

Calculate the side lengths

Using the Distance Formula knowledge point
\[

$$\begin{aligned} &IH = \sqrt{(-3 - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 9} = 3\\ &IJ = \sqrt{(0 - (-3))^2 + (4 - 4)^2} = \sqrt{9 + 0} = 3\\ &JK = \sqrt{(2 - 0)^2 + (-1 - 4)^2} = \sqrt{4 + 25} = \sqrt{29}\\ &HK = \sqrt{(2 - (-3))^2 + (-1 - 1)^2} = \sqrt{25 + 4} = \sqrt{29} \end{aligned}$$

\]

Verify the conditions for a kite

Using the Kite Properties knowledge point
\[

$$\begin{aligned} &IH = IJ = 3\\ &JK = HK = \sqrt{29}\\ &IH eq JK \text{ and } IJ eq HK \end{aligned}$$

\]
These calculations prove that \(HIJK\) has two distinct pairs of congruent adjacent sides, which matches the second option.
</reasoning>

<answer>
<mcq-option>(A) \(\overline{HI} \perp \overline{IJ}\), and \(m\angle H = m\angle J\).</mcq-option>
<mcq-correct>(B) \(IH = IJ = 3\) and \(JK = HK = \sqrt{29}\), and \(IH
eq JK\) and \(IJ
eq HK\).</mcq-correct>
<mcq-option>(C) \(\overline{IK}\) intersects \(\overline{HJ}\) at the midpoint of \(\overline{HJ}\) at \((-1.5, 2.5)\).</mcq-option>
<mcq-option>(D) The slope of \(\overline{HK} = -\frac{2}{5}\) and the slope of \(\overline{JK} = -\frac{5}{2}\).</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the definition of a kite

Using the Kite Properties knowledge point
A kite is defined as a quadrilateral with two distinct pairs of adjacent congruent sides. For quadrilateral \(HIJK\), this means we must show that one pair of adjacent sides is congruent (e.g., \(IH = IJ\)) and the other pair of adjacent sides is congruent (e.g., \(JK = HK\)), while ensuring that opposite sides are not congruent (\(IH
eq JK\) and \(IJ
eq HK\)) to distinguish it from a rhombus.

Find the coordinates of the vertices

We read the coordinates of the vertices from the given coordinate plane:

• \(H = (-3, 1)\)
• \(I = (-3, 4)\)
• \(J = (0, 4)\)
• \(K = (2, -1)\)

Calculate the side lengths

Using the Distance Formula knowledge point
\[

$$\begin{aligned} &IH = \sqrt{(-3 - (-3))^2 + (4 - 1)^2} = \sqrt{0 + 9} = 3\\ &IJ = \sqrt{(0 - (-3))^2 + (4 - 4)^2} = \sqrt{9 + 0} = 3\\ &JK = \sqrt{(2 - 0)^2 + (-1 - 4)^2} = \sqrt{4 + 25} = \sqrt{29}\\ &HK = \sqrt{(2 - (-3))^2 + (-1 - 1)^2} = \sqrt{25 + 4} = \sqrt{29} \end{aligned}$$

\]

Verify the conditions for a kite

Using the Kite Properties knowledge point
\[

$$\begin{aligned} &IH = IJ = 3\\ &JK = HK = \sqrt{29}\\ &IH eq JK \text{ and } IJ eq HK \end{aligned}$$

\]
These calculations prove that \(HIJK\) has two distinct pairs of congruent adjacent sides, which matches the second option.
</reasoning>

<answer>
<mcq-option>(A) \(\overline{HI} \perp \overline{IJ}\), and \(m\angle H = m\angle J\).</mcq-option>
<mcq-correct>(B) \(IH = IJ = 3\) and \(JK = HK = \sqrt{29}\), and \(IH
eq JK\) and \(IJ
eq HK\).</mcq-correct>
<mcq-option>(C) \(\overline{IK}\) intersects \(\overline{HJ}\) at the midpoint of \(\overline{HJ}\) at \((-1.5, 2.5)\).</mcq-option>
<mcq-option>(D) The slope of \(\overline{HK} = -\frac{2}{5}\) and the slope of \(\overline{JK} = -\frac{5}{2}\).</mcq-option>
</answer>

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