Question

is trapezoid abdc the result of a dilation of trapezoid mnpq by a scale factor of \\( \frac { 2 } { 5 } \\) ? why or why not?
yes, because ab and cd are each \\( \frac { 2 } { 5 } \\) the lengths mn and qp.
yes, because sides ab and cd are parallel to sides mn and qp.
no, because ab is \\( \frac { 2 } { 5 } \\) the length mn but cd is \\( \frac { 1 } { 3 } \\) the length qp.
no, because sides ab and cd have different slopes from sides mn and qp.

Explanation:

Step1: Check the scale - factor for corresponding sides

For a dilation, all corresponding sides must be scaled by the same factor.
Let's assume \(MN\) has length \(x\) and \(QP\) has length \(y\).
If \(AB=\frac{2}{5}x\) (assuming a scale - factor of \(\frac{2}{5}\) for \(AB\) with respect to \(MN\)) and \(CD = \frac{1}{3}y\) (not \(\frac{2}{5}y\) if the scale - factor is \(\frac{2}{5}\)).
Since the scale - factors for \(AB\) (with respect to \(MN\)) and \(CD\) (with respect to \(QP\)) are not the same (\(\frac{2}{5}
eq\frac{1}{3}\)), trapezoid \(ABDC\) is not a dilation of trapezoid \(MNPQ\) with a scale - factor of \(\frac{2}{5}\).

Step2: Analyze other options

• Option 1: Just because \(AB\) and \(CD\) are fractions of \(MN\) and \(QP\) (but not the same fraction) does not make it a dilation.
• Option 2: Parallelism of some sides (\(AB\parallel MN\) and \(CD\parallel QP\)) is a property of trapezoids but not sufficient for a dilation (which requires uniform scaling of all corresponding sides).
• Option 4: In a dilation, the slopes of corresponding sides are the same (since dilation is a similarity transformation that preserves the shape, including the orientation of sides). But the non - uniform scaling (shown in Option 3) is a more fundamental reason for it not being a dilation.

Answer:

No, because AB is $\frac{2}{5}$ the length MN but CD is $\frac{1}{3}$ the length QP.