Question

determine whether triangle mno is a dilation of triangle mno. the transformation is
○ an enlargement with scale factor 2.
○ a reduction with scale factor (\frac{1}{9}).
○ a reduction with scale factor (\frac{1}{2}).
○ not a dilation because the points of the image are not moved toward the center of dilation proportionally.

Explanation:

Step1: Recall dilation definition

Dilation requires proportional distances from center \( C \) for corresponding points. Check ratios of \( CM'/CM \), \( CN'/CN \), \( CO'/CO \).

Step2: Calculate \( CM'/CM \)

\( CM' = 3 \), \( CM = 3 + 3 = 6 \). Ratio: \( \frac{3}{6} = \frac{1}{2} \).

Step3: Calculate \( CN'/CN \)

\( CN' = 3 \), \( CN = 3 + 6 = 9 \)? Wait, no—wait, the segment from \( C \) to \( N' \) is 3, and from \( N' \) to \( N \) is 6? Wait, no, looking at the diagram: \( CN' \) is 3, \( N'N \) is 6? Wait, no, the horizontal line: \( C \) to \( N' \) is 3, \( N' \) to \( N \) is 6? Wait, no, actually, \( CN = CN' + N'N = 3 + 6 = 9 \)? Wait, no, maybe I misread. Wait, the length from \( C \) to \( M \) is \( 3 + 3 = 6 \) (since \( CM' = 3 \), \( M'M = 3 \)). From \( C \) to \( O \): \( CO' = 2 \), \( O'O = 6 \), so \( CO = 2 + 6 = 8 \)? Wait, no, that can't be. Wait, no—wait, the problem is about proportionality. Let's check each ratio:

• For \( M \): \( CM' = 3 \), \( CM = 3 + 3 = 6 \). Ratio \( \frac{CM'}{CM} = \frac{3}{6} = \frac{1}{2} \).
• For \( N \): Let's see, the segment from \( C \) to \( N' \) is 3, and from \( C \) to \( N \) is \( 3 + 6 = 9 \)? Wait, no, maybe the horizontal segment: \( N' \) to \( N \) is 6, so \( CN = CN' + N'N = 3 + 6 = 9 \)? Then \( \frac{CN'}{CN} = \frac{3}{9} = \frac{1}{3} \). Wait, that's not \( \frac{1}{2} \). Wait, maybe I misread the diagram. Wait, the vertical? No, the lines are from \( C \) to \( M \), \( C \) to \( N \), \( C \) to \( O \). Let's check \( O \): \( CO' = 2 \), \( CO = 2 + 6 = 8 \)? No, that's not. Wait, no—wait, the problem says "not a dilation because the points of the image are not moved toward the center of dilation proportionally." Let's check the ratios:

• \( CM' = 3 \), \( CM = 3 + 3 = 6 \) → ratio \( \frac{1}{2} \)
• \( CN' \): Let's see, the distance from \( C \) to \( N' \) is 3, and from \( C \) to \( N \) is \( 3 + 6 = 9 \)? Wait, no, maybe the segment from \( N' \) to \( N \) is 6, so \( CN = 3 + 6 = 9 \), so \( \frac{CN'}{CN} = \frac{3}{9} = \frac{1}{3} \)
• \( CO' = 2 \), \( CO = 2 + 6 = 8 \)? No, that's not. Wait, no—wait, the lengths from \( C \) to \( M' \) is 3, \( C \) to \( M \) is 6 (3+3). From \( C \) to \( O' \) is 2, \( C \) to \( O \) is 8 (2+6)? No, that can't be. Wait, maybe the diagram has \( CO' = 2 \), \( OO' = 6 \)? No, \( O' \) is between \( C \) and \( O \), so \( CO = CO' + OO' = 2 + 6 = 8 \). \( CM = CM' + MM' = 3 + 3 = 6 \). \( CN = CN' + NN' = 3 + 6 = 9 \). So the ratios are \( \frac{3}{6} = \frac{1}{2} \), \( \frac{3}{9} = \frac{1}{3} \), \( \frac{2}{8} = \frac{1}{4} \). These ratios are not equal, so the transformation is not a dilation because the points are not moved toward \( C \) proportionally.

Answer:

not a dilation because the points of the image are not moved toward the center of dilation proportionally.