Question

triangle tuv was dilated to create triangle tuv using point a as the center of dilation.
what is the scale factor of the dilation?
options: \frac{3}{2}, \frac{2}{5}, \frac{2}{3}, \frac{8}{5}

Explanation:

Step1: Recall dilation scale factor

The scale factor of dilation is the ratio of the length of a side of the image (dilated triangle) to the length of the corresponding side of the original triangle. Here, we can use the segments from the center of dilation \( A \) to the vertices. The length from \( A \) to \( V' \) is \( 3.2 \) and from \( A \) to \( V \) is \( 3.2 + 4.8 = 8 \)? Wait, no, actually, the scale factor is \( \frac{\text{length of } AV'}{\text{length of } AV} \)? Wait, no, wait: the dilated triangle is \( T'U'V' \) and original is \( TUV \). So the distance from \( A \) to \( V' \) is \( 3.2 \), and from \( A \) to \( V \) is \( 3.2 + 4.8 = 8 \)? Wait, no, maybe I got it reversed. Wait, dilation: if \( T'U'V' \) is the image, then the scale factor \( k = \frac{AV'}{AV} \)? Wait, no, let's check the lengths. \( AV' = 3.2 \), \( AV = 3.2 + 4.8 = 8 \)? Wait, no, \( AV' \) is the distance from \( A \) to \( V' \), and \( AV \) is from \( A \) to \( V \). So the scale factor is \( \frac{AV'}{AV} \)? Wait, no, actually, when you dilate, the scale factor is \( \frac{\text{length of image segment}}{\text{length of original segment}} \). So if \( V' \) is on \( AV \), then \( AV' \) is the image segment, \( AV \) is the original? Wait, no, the original triangle is \( TUV \), and the dilated is \( T'U'V' \), so the center is \( A \), so the vectors \( \overrightarrow{AT'} = k \overrightarrow{AT} \), \( \overrightarrow{AU'} = k \overrightarrow{AU} \), \( \overrightarrow{AV'} = k \overrightarrow{AV} \). So the length \( AV' = k \times AV \), so \( k = \frac{AV'}{AV} \). Wait, \( AV' = 3.2 \), \( AV = AV' + V'V = 3.2 + 4.8 = 8 \). So \( k = \frac{3.2}{8} = 0.4 = \frac{2}{5} \)? Wait, no, that can't be. Wait, maybe I mixed up image and original. Wait, the problem says "Triangle TUV was dilated to create triangle T'U'V'". So TUV is original, T'U'V' is image. So the scale factor is \( \frac{\text{length of } AT'}{\text{length of } AT} \), or \( \frac{AU'}{AU} \), or \( \frac{AV'}{AV} \). Let's compute \( AV' = 3.2 \), \( AV = 3.2 + 4.8 = 8 \). So \( \frac{AV'}{AV} = \frac{3.2}{8} = 0.4 = \frac{2}{5} \)? Wait, but let's check the options. The options are \( \frac{3}{2} \), \( \frac{2}{5} \), \( \frac{2}{3} \), \( \frac{8}{5} \). Wait, maybe I got the direction wrong. Wait, maybe \( AV \) is \( 3.2 \), and \( AV' \) is \( 3.2 + 4.8 = 8 \)? No, that would be if \( V' \) is beyond \( V \), but in the diagram, \( V' \) is between \( A \) and \( V \), so \( AV' = 3.2 \), \( VV' = 4.8 \), so \( AV = AV' + VV' = 3.2 + 4.8 = 8 \). So the scale factor is \( \frac{AV'}{AV} = \frac{3.2}{8} = 0.4 = \frac{2}{5} \). Wait, but let's confirm. Dilation scale factor: if the image is smaller than the original, the scale factor is less than 1. Here, triangle T'U'V' is smaller than TUV, so scale factor should be less than 1. \( \frac{2}{5} = 0.4 \), which is less than 1. Let's check the calculation again. \( 3.2 \div (3.2 + 4.8) = 3.2 \div 8 = 0.4 = \frac{2}{5} \). Yes, that's correct.

Step2: Calculate the scale factor

We have \( AV' = 3.2 \) and \( AV = 3.2 + 4.8 = 8 \). The scale factor \( k \) is given by \( k = \frac{AV'}{AV} \). Substituting the values, we get \( k = \frac{3.2}{8} = \frac{32}{80} = \frac{2}{5} \).

Answer:

\(\frac{2}{5}\) (corresponding to the option with \(\frac{2}{5}\))