Question

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

Explanation:

Step1: Recall Dilation Properties

Dilation preserves parallelism and scales all corresponding linear measurements by the same scale factor. For trapezoids, the two parallel sides (bases) should be scaled by the same factor, and non - parallel sides (legs) should also be scaled by the same factor.

Step2: Analyze Each Option

• Option 1: Check the scale factor of the bases. Let's assume the length of \(MN\) (base of the larger trapezoid) and \(QP\) (the other base), and \(AB\) and \(CD\) (bases of the smaller trapezoid). If we calculate the ratio of \(AB\) to \(MN\) and \(CD\) to \(QP\), we need to see if they are both \(\frac{2}{5}\). From the graph (assuming grid - based calculation), if \(MN\) has a length (say, number of grid units) and \(AB\) is a part of it, and similarly for \(CD\) and \(QP\), if \(AB=\frac{2}{5}MN\) and \(CD = \frac{2}{5}QP\), then the scale factor is consistent for the parallel sides. Also, since dilation preserves parallelism, the parallel sides remain parallel.
• Option 2: Just stating that \(AB\) and \(CD\) are parallel to \(MN\) and \(QP\) (which is true for trapezoids) is not enough to confirm dilation. Dilation requires scaling by the same factor.
• Option 3: If \(AB=\frac{2}{5}MN\) but \(CD=\frac{1}{3}QP\), the scale factors are different, so it's not a dilation. But this is not the case here.
• Option 4: In a dilation, the slopes of corresponding sides should be the same (since dilation is a similarity transformation that preserves the shape, so the angle - related properties like slope for non - vertical/horizontal lines are preserved). So saying they have different slopes is incorrect.

Answer:

A. Yes, because AB and CD are each \(\frac{2}{5}\) the lengths MN and QP.