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.
no, because sides ab and cd have different slopes from sides mn and qp.
no, because ab is \\(\frac{2}{5}\\) the length mn but cd is \\(\frac{1}{3}\\) the length qp.
yes, because sides ab and cd are parallel to sides mn and qp.

Explanation:

Step1: Determine lengths of MN, QP, AB, CD

First, find the length of \( MN \). From the graph, \( M \) is at \( x = -5 \) (assuming grid units, since from -5 to 5? Wait, looking at the grid, \( M \) is at \( x = -5 \)? Wait, the grid has M at left, N at right. Let's count the units. \( MN \): from \( x = -5 \) to \( x = 5 \)? Wait, no, the coordinates: M is at (-5, 4), N at (5, 4)? Wait, no, the graph shows M at left, N at right, with MN horizontal. Let's check the length of MN: from x=-5 to x=5? Wait, no, the grid lines: each square is 1 unit. So M is at (-5, 4), N at (5, 4), so length \( MN = 5 - (-5) = 10 \)? Wait, no, looking at the smaller trapezoid AB: A is at (-2, 2), B at (2, 2), so AB length is \( 2 - (-2) = 4 \)? Wait, no, the first option says AB is \( \frac{2}{5} \) of MN. Wait, maybe I miscalculate. Let's re-express:

Looking at the graph:

• \( MN \): from x = -5 to x = 5? Wait, no, the top base of the larger trapezoid MNPQ: M is at (-5, 4), N at (5, 4), so length \( MN = 10 \) (from -5 to 5, 10 units).

• \( AB \): top base of smaller trapezoid ABDC: A at (-2, 2), B at (2, 2), so length \( AB = 4 \) (from -2 to 2, 4 units). Wait, \( \frac{4}{10} = \frac{2}{5} \), that works.

• \( QP \): bottom base of larger trapezoid MNPQ: Q at (-3, -2), P at (3, -2)? Wait, no, Q is at (-3, -2), P at (3, -2)? Wait, no, the bottom base QP: from x = -3 to x = 3? Wait, no, looking at the graph, Q is at (-3, -2), P at (3, -2), so length \( QP = 6 \) (from -3 to 3, 6 units).

• \( CD \): bottom base of smaller trapezoid ABDC: C at (-1, 0), D at (1, 0), so length \( CD = 2 \) (from -1 to 1, 2 units). Wait, \( \frac{2}{6} = \frac{1}{3} \), not \( \frac{2}{5} \). Ah! So AB is \( \frac{4}{10} = \frac{2}{5} \) (if MN is 10), but CD is \( \frac{2}{6} = \frac{1}{3} \) (if QP is 6). So AB is \( \frac{2}{5} \) of MN, but CD is \( \frac{1}{3} \) of QP. Therefore, the scale factor is not consistent. So the third option: "No, because AB is \( \frac{2}{5} \) the length MN but CD is \( \frac{1}{3} \) the length QP." is correct.

Step2: Analyze each option

• First option: "Yes, because AB and CD are each \( \frac{2}{5} \) the lengths MN and QP." But we saw CD is \( \frac{1}{3} \) of QP, not \( \frac{2}{5} \), so wrong.
• Second option: "No, because sides AB and CD have different slopes from sides MN and QP." But all are horizontal (slope 0), so slopes are same. Wrong.
• Third option: "No, because AB is \( \frac{2}{5} \) the length MN but CD is \( \frac{1}{3} \) the length QP." This matches our calculation: AB length / MN length = 4/10 = 2/5, CD length / QP length = 2/6 = 1/3. So scale factor not consistent, so dilation (which requires all corresponding sides to be scaled by same factor) is not done. Correct.
• Fourth option: "Yes, because sides AB and CD are parallel to sides MN and QP." Parallelism is necessary for trapezoid, but dilation requires scale factor consistency, not just parallelism. So wrong.

Answer:

No, because AB is \( \frac{2}{5} \) the length MN but CD is \( \frac{1}{3} \) the length QP. (The third option in the list)