Question

which statement proves that quadrilateral jklm is a kite?

• ( angle m ) is a right angle and ( overline{mk} ) bisects ( angle lmj ).
• ( lm = jm = 3 ) and ( jk = lk = sqrt{17} ).
• ( overline{mk} ) intersects ( overline{lj} ) at its midpoint.
• the slope of ( overline{mk} ) is ( -1 ) and the slope of ( overline{lj} ) is ( 1 ).

Explanation:

Step1: Recall kite definition

A kite is a quadrilateral with two distinct pairs of adjacent sides equal.

Step2: Calculate side lengths

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

• $LM$: $\sqrt{(1-1)^2+(5-2)^2}=\sqrt{0+9}=3$
• $JM$: $\sqrt{(4-1)^2+(5-5)^2}=\sqrt{9+0}=3$
• $JK$: $\sqrt{(5-4)^2+(1-5)^2}=\sqrt{1+16}=\sqrt{17}$
• $LK$: $\sqrt{(5-1)^2+(1-2)^2}=\sqrt{16+1}=\sqrt{17}$

Step3: Match to kite property

This gives two pairs of adjacent equal sides: $LM=JM$ and $JK=LK$, which satisfies the kite definition.

Step4: Eliminate other options

• Right angle + angle bisector does not prove kite properties.
• Diagonal intersecting midpoint describes a parallelogram, not a kite.
• Perpendicular slopes mean diagonals are perpendicular, which is true for kites but does not uniquely prove it (e.g., rhombuses also have perpendicular diagonals).

Answer:

LM = JM = 3 and JK = LK = $\sqrt{17}$