Question

the pink figure is a dilation image of the black figure. the labeled point is the center of dilation. tell whether the dilation is an enlargement or a reduction. then find the scale factor of the dilation

reduction
enlargement
the scale factor of the dilation is \\(\square\\) (simplify your answer.)

Explanation:

Step1: Determine Dilation Type

Compare the size of the pink (image) and black (pre - image) figures. The pink figure is larger, so it's an enlargement.

Step2: Find Scale Factor

To find the scale factor, we can look at the side lengths. Let's assume the side length of a small square in the black figure is 1 unit, and in the pink figure, the corresponding side length is 3 units (by counting the grid squares). The scale factor \(k\) is the ratio of the length of the image to the length of the pre - image. If we take a corresponding side, say the side of the small square - like part, in the black figure it might be 1 unit and in the pink figure it's 3 units? Wait, no, actually, looking at the figure, the black figure has a smaller size and the pink is larger. Let's take a segment. Suppose in the pre - image (black) a side is 1 unit, and in the image (pink) the corresponding side is 3 units? Wait, no, maybe the other way. Wait, actually, let's count the number of grid squares. Let's say the black figure's small part has a side length of 1 (like the small square in the black), and the pink figure's corresponding part has a side length of 3? No, wait, maybe the scale factor is 3? Wait, no, let's think again. Wait, the center of dilation is the labeled point. Let's take a point on the black figure and its image on the pink figure. The distance from the center of dilation to a point on the black figure and to its image on the pink figure. If the distance from the center to a black point is \(d\) and to the pink point is \(3d\), then the scale factor is 3. Wait, maybe the black figure is smaller, and the pink is larger, so scale factor is 3. Wait, let's check: if the black figure has a side length of 1 (in terms of grid squares) and the pink has 3, then scale factor \(k=\frac{\text{length of image}}{\text{length of pre - image}}=\frac{3}{1} = 3\)? Wait, no, maybe I got it reversed. Wait, no, dilation: if the image is larger, scale factor \(k>1\) (enlargement), if smaller, \(k < 1\) (reduction). Since it's an enlargement, \(k>1\). Let's take a specific example. Suppose in the black figure, a side of a small square is 1 unit, and in the pink figure, the corresponding side is 3 units. So scale factor \(k = 3\).

Answer:

Enlargement
The scale factor of the dilation is \(\boldsymbol{3}\)