Question

what is the scale factor of the dilation of line segment ba? options: 4, 5, \\(\frac{1}{5}\\), \\(\frac{1}{4}\\) (accompanying diagram of triangle with segments ca = 4, ca = 16, and segments ba, ba)

Explanation:

Step1: Recall dilation scale factor formula

The scale factor \( k \) of a dilation is given by the ratio of the length of the image segment to the length of the original segment. For dilation, \( k=\frac{\text{Length of image segment}}{\text{Length of original segment}} \).

Step2: Identify original and image lengths

From the diagram, the original segment \( BA \) is part of the smaller triangle, and the image segment \( B'A' \) is part of the larger triangle. The length from \( C \) to \( A \) is \( 4 \), and from \( C \) to \( A' \) is \( 16 \). Since dilation is centered at \( C \), the scale factor for the segments from \( C \) will be the same as for \( BA \) and \( B'A' \).

So, the length of the original segment (from \( C \) to \( A \)) is \( 4 \), and the length of the image segment (from \( C \) to \( A' \)) is \( 16 \).

Step3: Calculate the scale factor

Using the scale factor formula, \( k = \frac{\text{Length of } CA'}{\text{Length of } CA}=\frac{16}{4} = 4 \)? Wait, no, wait. Wait, actually, if \( BA \) is the original and \( B'A' \) is the image, but let's check again. Wait, maybe I mixed up. Wait, the original segment is \( BA \), and the image is \( B'A' \). But also, the segment from \( C \) to \( A \) is \( 4 \), and from \( C \) to \( A' \) is \( 16 \). So the scale factor is \( \frac{CA'}{CA}=\frac{16}{4} = 4 \)? Wait, but let's confirm. Dilation scale factor is image over pre - image. So if \( CA \) is pre - image (length 4) and \( CA' \) is image (length 16), then scale factor \( k=\frac{16}{4}=4 \). So the scale factor for \( BA \) to \( B'A' \) is also 4, because dilation preserves the shape, so the ratio of corresponding segments is the same.

Answer:

\( 4 \) (corresponding to the option with value 4)