Question

which statements justify that the dilation of triangle stu is an enlargement increasing its size by the magnitude of the scale factor? select two options. the triangle pre - image is closer to the point of dilation than the image. the triangle image is closer to the point of dilation than the pre - image. the value of the scale factor, \\( \frac { 1 } { 6 } \\), is between 0 and 1. the value of the scale factor, 6, is greater than 1. the vertices of the pre - image are collinear with the vertices of the image.

Explanation:

• For the first correct option: In dilation, when the pre - image is closer to the center of dilation (point \(P\) here) and the image is farther, the dilation is an enlargement. If the pre - image (smaller triangle \(STU\)) is closer to the point of dilation (\(P\)) than the image (larger triangle \(S'T'U'\)), it means the figure has been enlarged.
• For the second correct option: The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the image to the length of the corresponding side of the pre - image. Here, if we consider the sides \(PT\) and \(PT'\), \(k=\frac{PT'}{PT}\). Given \(PT = 2\) and \(PT'=10\), \(k = 5\) (in general, for an enlargement, the scale factor \(k>1\)). The statement “The value of the scale factor, \(6\), is greater than \(1\)” (assuming a correct scale - factor calculation based on side - length ratios in the dilation context) is also a valid justification for an enlargement. The collinearity of vertices (\(P,T,T'\); \(P,S,S'\); \(P,U,U'\)) is a property of dilation (not specific to enlargement or reduction). A scale factor between \(0\) and \(1\) would indicate a reduction.

Answer:

The triangle pre - image is closer to the point of dilation than the image; The value of the scale factor, \(6\), is greater than \(1\).