Question

is quadrilateral jklm the result of a dilation of quadrilateral abcd by a scale factor of 2? why or why not?
yes, because sides jk and ml are twice as long as sides ab and dc.
yes, because both figures are parallelograms, so corresponding sides are parallel.
no, because sides jk and ml are not twice as long as sides ab and dc.
no, because sides jm and kl have different slopes from sides ad and bc.

Explanation:

Step1: Recall the property of dilation

Dilation is a transformation that changes the size of a figure by a scale factor. If a figure is dilated by a scale factor \(k\), then all of its corresponding sides are multiplied by \(k\).

Step2: Calculate the length of sides \(AB\) and \(JK\)

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

Step3: Calculate the length of sides \(DC\) and \(ML\)

For side \(DC\) with \(D(1,-4)\) and \(C(3,-4)\), \(DC=\sqrt{(3 - 1)^2+(-4+4)^2}=2\). For side \(ML\) with \(M(3,-8)\) and \(L(7,-8)\), \(ML=\sqrt{(7 - 3)^2+(-8 + 8)^2}=4\).

Step4: Check other sides

For side \(AD\) with \(A(2,-2)\) and \(D(1,-4)\), \(AD=\sqrt{(1 - 2)^2+(-4 + 2)^2}=\sqrt{1 + 4}=\sqrt{5}\). For side \(JM\) with \(J(4,-4)\) and \(M(3,-8)\), \(JM=\sqrt{(3 - 4)^2+(-8 + 4)^2}=\sqrt{1+16}=\sqrt{17}
eq2\sqrt{5}\)

Answer:

No, because sides \(JK\) and \(ML\) are not twice as long as sides \(AB\) and \(DC\) (also, not all corresponding sides are in the ratio of \(2\)). So the correct option is “No, because sides \(JK\) and \(ML\) are not twice as long as sides \(AB\) and \(DC\)”