Question

identify the dilation as a reduction or enlargement, then identify the scale factor.

Explanation:

Step1: Identify the dilation type

A dilation is a reduction if the scale factor \( k \) satisfies \( 0 < k < 1 \), and an enlargement if \( k>1 \). We compare the lengths from the center of dilation \( C \). Let the length from \( C \) to the smaller figure be \( 5 \) and to the larger figure be \( 14 \). Wait, actually, for similar figures under dilation, the scale factor \( k=\frac{\text{length of image}}{\text{length of pre - image}} \). If the smaller triangle is the pre - image and the larger one is the image, or vice - versa. Let's assume the segment from \( C \) to a vertex of the smaller triangle is \( 5 \) and to the corresponding vertex of the larger triangle is \( 14 \). Wait, no, maybe the two segments from \( C \) are \( 5 \) (to the smaller figure) and \( 14 \) (to the larger figure). Wait, actually, the scale factor \( k=\frac{\text{distance from center to image vertex}}{\text{distance from center to pre - image vertex}} \). If the smaller figure is the pre - image and the larger is the image, then \( k = \frac{14}{5}=2.8>1 \), so it's an enlargement. Wait, maybe I got the pre - image and image reversed. Wait, the problem says "identify the dilation as a reduction or enlargement, then identify the scale factor". Let's assume that the two triangles are similar, with the distance from \( C \) to the smaller triangle's vertex \( P \) (let's say) is \( 5 \) and to the larger triangle's vertex \( P' \) is \( 14 \). So the scale factor \( k=\frac{CP'}{CP}=\frac{14}{5} = 2.8\) or \( \frac{5}{14}\approx0.357 \). Wait, but if the smaller triangle is the image and the larger is the pre - image, then it's a reduction with \( k=\frac{5}{14}\), but if the larger is the image and the smaller is the pre - image, it's an enlargement with \( k = \frac{14}{5}\). From the diagram, the larger triangle is the image (since it's bigger), so the dilation is an enlargement.

Step2: Calculate the scale factor

The scale factor \( k\) is the ratio of the corresponding lengths from the center of dilation. Let the length from \( C \) to the vertex of the larger triangle be \( 14 \) and to the vertex of the smaller triangle be \( 5 \). So \( k=\frac{14}{5}=2.8=\frac{14}{5}\).

Answer:

The dilation is an enlargement with a scale factor of \(\frac{14}{5}\) (or \(2.8\)).