Question

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

Explanation:

Step1: Analyze MN and AB lengths

MN spans from \( x = -4 \) to \( x = 4 \), so length \( MN = 4 - (-4) = 8 \). AB spans from \( x = -2 \) to \( x = 2 \), so length \( AB = 2 - (-2) = 4 \). The ratio \( \frac{AB}{MN} = \frac{4}{8} = \frac{2}{5} \).

Step2: Analyze QP and CD lengths

QP spans from \( x = -2 \) to \( x = 2 \)? Wait, no, Q is at \( x = -2 \)? Wait, looking at the grid, Q is at \( x = -3 \)? Wait, no, the lower base QP: Q is at \( x = -2 \)? Wait, no, let's recheck. Wait, the lower trapezoid MNPQ: M is at (-4, y), N at (4, y), Q at (-2, -3) (approx), P at (2, -3)? Wait, no, CD is from \( x = -1 \) to \( x = 1 \)? Wait, no, the inner trapezoid AB is from -2 to 2, CD is from -1 to 1? Wait, no, the x-axis: C is at -1, D at 1? Wait, no, the original trapezoid MNPQ: the top base MN is from -4 to 4 (length 8), bottom base QP: Q is at -2, P at 2? No, that can't be. Wait, maybe Q is at -3, P at 3? No, the grid lines: each square is 1 unit. Let's count the units for QP: Q is at x = -2, P at x = 2? No, the lower base QP: from x = -2 to x = 2? Then length QP = 4? No, that's not. Wait, maybe I misread. Wait, the inner trapezoid AB: A is at (-2, y), B at (2, y), so AB length is 4. CD: C at (-1, 0), D at (1, 0), so CD length is 2. Now QP: Q at (-3, -3), P at (3, -3)? No, the lower base MNPQ: M at (-4, 1), N at (4, 1), Q at (-2, -3), P at (2, -3)? Wait, no, the x-coordinates: M is at -4, N at 4 (length 8), Q at -2, P at 2 (length 4)? No, that would make QP length 4. Then CD length is 2, so \( \frac{CD}{QP} = \frac{2}{6} \)? Wait, no, maybe QP is from -3 to 3? Wait, I think I made a mistake. Let's do it correctly:

Top base MN: from \( x = -4 \) to \( x = 4 \), so length \( MN = 8 \) (since 4 - (-4) = 8).

AB: from \( x = -2 \) to \( x = 2 \), length \( AB = 4 \) (2 - (-2) = 4). So \( \frac{AB}{MN} = \frac{4}{8} = \frac{2}{5} \), correct.

Now QP: the bottom base of MNPQ. Let's see the lower vertices: Q and P. Looking at the grid, Q is at \( x = -2 \), P at \( x = 2 \)? No, that's length 4. But CD: the bottom base of ABDC is from \( x = -1 \) to \( x = 1 \), length 2. Wait, no, CD is from \( x = -1 \) to \( x = 1 \), so length 2. QP: if Q is at \( x = -3 \), P at \( x = 3 \), length 6. Then \( \frac{CD}{QP} = \frac{2}{6} = \frac{1}{3} \). Ah, that makes sense. So QP length is 6 (from -3 to 3), CD length is 2. So \( \frac{CD}{QP} = \frac{2}{6} = \frac{1}{3} \), which is not \( \frac{2}{5} \).

Step3: Evaluate the options

Option 1: Says AB and CD are each \( \frac{2}{5} \) of MN and QP. But CD is \( \frac{1}{3} \) (if QP is 6) or \( \frac{1}{2} \) (if QP is 4). Wait, earlier mistake: let's recalculate QP. If M is at (-4, y), N at (4, y) (length 8), Q is at (-3, -y), P at (3, -y) (length 6). Then QP length is 6. CD is from -1 to 1 (length 2). So \( \frac{CD}{QP} = \frac{2}{6} = \frac{1}{3} \), and \( \frac{AB}{MN} = \frac{4}{8} = \frac{2}{5} \). So AB is \( \frac{2}{5} \) of MN, but CD is \( \frac{1}{3} \) of QP. So option 3 says "No, because AB is \( \frac{2}{5} \) the length MN but CD is \( \frac{1}{3} \) the length QP." That matches.

Answer:

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