Question

which statement proves that quadrilateral hijk is a kite? ∘ $overline{hi} perp overline{ij}$, and $mangle h = mangle j$. ∘ $ih = ij = 3$ 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:

A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal in length. Let's analyze each option:

• Option 1: \( \overline{HI} \perp \overline{IJ} \) (perpendicular sides) and \( m\angle H = m\angle J \) (equal angles) do not prove it's a kite, as a kite's definition is about side lengths, not angles or perpendicularity of adjacent sides.
• Option 2: \( IH = IJ = 3 \) (one pair of adjacent equal sides) and \( JK = HK=\sqrt{29} \) (another pair of adjacent equal sides), and \( IH

eq JK \), \( IJ
eq HK \) (so the pairs are distinct). This matches the definition of a kite (two distinct pairs of adjacent equal sides).

• Option 3: A kite's diagonals do not necessarily bisect each other (that's a property of parallelograms, and a kite is not a parallelogram unless it's a rhombus). So this does not prove it's a kite.
• Option 4: The slopes of \( \overline{HK} \) and \( \overline{JK} \) being \( -\frac{2}{5} \) and \( -\frac{5}{2} \) (negative reciprocals would mean perpendicular, but here they are not negative reciprocals, and slope information alone doesn't prove it's a kite).

Answer:

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