Question

which statement proves that quadrilateral j k l m is a kite?
∠m is a right angle and (overline{m k}) bisects ∠l m j.
l m = j m = 3 and j k = l k = (sqrt{17}).
(overline{m k}) intersects (overline{l j}) at its midpoint.
the slope of (overline{m k}) is - 1 and the slope of (overline{l j}) is 1.

Explanation:

Step1: Recall the definition of a kite

A kite is a quadrilateral with two distinct pairs of adjacent sides that are equal in length.

Step2: Calculate the lengths of the sides

• For \(LM\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(L(1,2)\) and \(M(1,5)\), \(x_1 = 1,x_2 = 1,y_1 = 2,y_2 = 5\). Then \(LM=\sqrt{(1 - 1)^2+(5 - 2)^2}=\sqrt{0 + 9}=3\).
• For \(JM\): For \(J(4,5)\) and \(M(1,5)\), \(x_1 = 1,x_2 = 4,y_1 = 5,y_2 = 5\). Then \(JM=\sqrt{(4 - 1)^2+(5 - 5)^2}=\sqrt{9+0}=3\).
• For \(JK\): For \(J(4,5)\) and \(K(5,1)\), \(x_1 = 4,x_2 = 5,y_1 = 5,y_2 = 1\). Then \(JK=\sqrt{(5 - 4)^2+(1 - 5)^2}=\sqrt{1 + 16}=\sqrt{17}\).
• For \(LK\): For \(L(1,2)\) and \(K(5,1)\), \(x_1 = 1,x_2 = 5,y_1 = 2,y_2 = 1\). Then \(LK=\sqrt{(5 - 1)^2+(1 - 2)^2}=\sqrt{16 + 1}=\sqrt{17}\).

Since \(LM = JM = 3\) and \(JK=LK=\sqrt{17}\), which satisfies the definition of a kite (two - distinct pairs of adjacent sides are equal).

Answer:

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