Question

how do you see it? point c is the center of dilation of the image. the scale factor is \\( \frac{1}{3} \\). which figure is the original figure? which figure is the dilated figure? explain your reasoning.

Explanation:

Step1: Recall Scale Factor Concept

A scale factor \( k \) in a dilation (with center \( C \)) relates the original figure (\( F \)) and the image (\( F' \)) by \( \text{length of } F' = k \times \text{length of } F \). If \( k > 1 \), the image is an enlargement; if \( 0 < k < 1 \), the image is a reduction.

Step2: Analyze the Stars' Sizes

The smaller star is above the larger star, with center \( C \) above the smaller star. The scale factor here is \( \frac{1}{3} \) (a reduction factor, since \( 0 < \frac{1}{3} < 1 \)). For a dilation with scale factor \( \frac{1}{3} \), the image is the smaller figure, and the original is the larger figure. Because when \( k=\frac{1}{3} \), \( \text{image length} = \frac{1}{3} \times \text{original length} \), so the original must be larger (since multiplying by a fraction less than 1 reduces the size). Thus, the larger star is the original figure, and the smaller star is the dilated (image) figure. The scale factor \( \frac{1}{3} \) means the image (small star) is \( \frac{1}{3} \) the size of the original (large star).

Answer:

• Scale Factor Interpretation: A scale factor of \( \frac{1}{3} \) (reduction) means the image is \( \frac{1}{3} \) the original’s size.
• Original Figure: The larger star (since the smaller star is the reduced image, as \( \frac{1}{3} < 1 \), so original must be larger to get a smaller image via dilation from center \( C \)).
• Dilated Figure: The smaller star (it is \( \frac{1}{3} \) the size of the larger star, formed by dilating the larger star with scale factor \( \frac{1}{3} \) from center \( C \)).