Question

which statement proves that quadrilateral jklm is a kite?
∠m is a right angle and mk bisects ∠lmj.
lm = jm = 3 and jk = lk = √17.
mk intersects lj at its midpoint.
the slope of mk is -1 and the slope of lj is 1.

Explanation:

Step1: Recall kite - definition

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

Step2: Calculate side - lengths

For two points \((x_1,y_1)\) and \((x_2,y_2)\), the distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(L(1,2)\) and \(M(1,5)\), \(LM=\sqrt{(1 - 1)^2+(5 - 2)^2}=\sqrt{0 + 9}=3\).
For \(J(4,5)\) and \(M(1,5)\), \(JM=\sqrt{(4 - 1)^2+(5 - 5)^2}=\sqrt{9+0}=3\).
For \(J(4,5)\) and \(K(5,1)\), \(JK=\sqrt{(5 - 4)^2+(1 - 5)^2}=\sqrt{1+( - 4)^2}=\sqrt{1 + 16}=\sqrt{17}\).
For \(L(1,2)\) and \(K(5,1)\), \(LK=\sqrt{(5 - 1)^2+(1 - 2)^2}=\sqrt{16+( - 1)^2}=\sqrt{16 + 1}=\sqrt{17}\).
Since \(LM = JM = 3\) and \(JK = LK=\sqrt{17}\), the quadrilateral \(JKLM\) is a kite.

Answer:

LM = JM = 3 and JK = LK = \(\sqrt{17}\).