Question

is quadrilateral j k l m the result of a dilation of quadrilateral a b c d by a scale factor of 2? why or why not?
yes, because sides j k and m l are twice as long as sides a b and d c.
yes, because both figures are parallelograms, so corresponding sides are parallel.
no, because sides j k and m l are not twice as long as sides a b and d c.
no, because sides j m and k l have different slopes from sides a d and b c.

Explanation:

Step1: Calculate the length of AB and JK

• For side AB: Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), if \(A=(2,-2)\) and \(B=(4,-2)\), then \(AB=\sqrt{(4 - 2)^2+(-2+ 2)^2}=2\)
• For side JK: If \(J=(4,-4)\) and \(K=(8,-4)\), then \(JK=\sqrt{(8 - 4)^2+(-4 + 4)^2}=4\)

Step2: Check the scale - factor for other sides

• For side DC: If \(D=(1,-4)\) and \(C=(3,-4)\), then \(DC = 2\)
• For side ML: If \(M=(3,-9)\) and \(L=(7,-9)\), then \(ML=4\)
• But for side AD: If \(A=(2,-2)\) and \(D=(1,-4)\), \(AD=\sqrt{(2 - 1)^2+(-2 + 4)^2}=\sqrt{1 + 4}=\sqrt{5}\)
• For side JM: If \(J=(4,-4)\) and \(M=(3,-9)\), \(JM=\sqrt{(4 - 3)^2+(-4 + 9)^2}=\sqrt{1+25}=\sqrt{26}

eq2\sqrt{5}\)

A dilation requires all corresponding sides to be in proportion. Since not all sides of \(JKLM\) are twice the length of the corresponding sides of \(ABCD\) (e.g., \(JM
eq2AD\)), it is not a dilation by a scale - factor of 2.

Answer:

No, because sides \(JK\) and \(ML\) are not twice as long as sides \(AB\) and \(DC\) (the third option).