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: Recall the property of dilation

Dilation is a transformation that changes the size of a figure but not its shape. All corresponding sides of the original and dilated figure are in proportion equal to the scale factor.

Step2: Calculate the length of sides

Let's assume the side - length calculation using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a parallelogram \(ABCD\) and \(JKLM\). If we consider the horizontal - like sides (by counting the grid units in the direction parallel to their length).
Let's say for side \(AB\) (in parallelogram \(ABCD\)): if we assume \(A=(2, - 1)\) and \(B=(4,-1)\), \(AB=\sqrt{(4 - 2)^2+(-1+1)^2}=2\) units. For side \(JK\) (in parallelogram \(JKLM\)): if \(J=(4,2)\) and \(K=(8,2)\), \(JK=\sqrt{(8 - 4)^2+(2 - 2)^2}=4\) units. But for non - horizontal sides: assume \(AD\) with \(A=(2,-1)\) and \(D=(1,1)\), \(AD=\sqrt{(1 - 2)^2+(1 + 1)^2}=\sqrt{1 + 4}=\sqrt{5}\). For \(JM\) with \(J=(4,2)\) and \(M=(2,5)\), \(JM=\sqrt{(2 - 4)^2+(5 - 2)^2}=\sqrt{4 + 9}=\sqrt{13}\). \(\frac{JM}{AD}
eq2\)

Answer:

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