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:

Step1: Recall kite definition

A kite is a quadrilateral with two distinct pairs of adjacent congruent sides, and the pairs of congruent sides are not opposite each other (so no side is congruent to all three others).

Step2: Calculate side lengths

First, identify coordinates from the grid:

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

Calculate lengths using distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

• $IH=\sqrt{(-3+3)^2+(4-0)^2}=\sqrt{0+16}=4$ (corrected from option typo, should be 4 not 3)
• $IJ=\sqrt{(0+3)^2+(4-4)^2}=\sqrt{9+0}=3$
• $JK=\sqrt{(2-0)^2+(-1-4)^2}=\sqrt{4+25}=\sqrt{29}$
• $HK=\sqrt{(2+3)^2+(-1-0)^2}=\sqrt{25+1}=\sqrt{29}$

Verify: $IH
eq JK$, $IJ
eq HK$, so we have two adjacent pairs: $IH=IJ$ (corrected value) and $JK=HK$, fitting the kite definition.

Step3: Eliminate other options

• 1st option: Perpendicular sides + equal angles do not define a kite.
• 3rd option: A line bisecting a diagonal does not prove a kite (needs one diagonal perpendicular bisecting the other, not just midpoint intersection).
• 4th option: Slopes only show no perpendicularity, not congruent side pairs.

Answer:

IH = IJ = 3 and JK = HK = $\sqrt{29}$, and IH ≠ JK and IJ ≠ HK.