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 $\overline{jk} = -\frac{5}{2}$.

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 $\overline{jk} = -\frac{5}{2}$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Kite Properties", "Distance Formula" ], "new_concepts": [ "Coordinate Proofs of Quadrilaterals" ], "current_concepts": [ "Kite Properties", "Distance Formula", "Coordinate Proofs of Quadrilaterals" ] } ## 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.

(A) $\overline{HI} \perp \overline{IJ}$, and $m\angle H = m\angle J$.

(B) $IH = IJ = 3$ and $JK = HK = \sqrt{29}$, and $IH eq JK$ and $IJ eq HK$.

(C) $\overline{IK}$ intersects $\overline{HJ}$ at the midpoint of $\overline{HJ}$ at $(-1.5, 2.5)$.

(D) The slope of $\overline{HK} = -\frac{2}{5}$ and the slope of $\overline{JK} = -\frac{5}{2}$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Proofs of Quadrilaterals" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the definition of a kite

Find the coordinates of the vertices

Calculate the side lengths

Verify the conditions for a kite

Answer: