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:

Step1: Understand dilation properties

Dilation is a transformation that enlarges or reduces a figure. If the scale factor \(k> 1\), it is an enlargement. Also, in dilation, the pre - image and image are similar and the points (vertices) of the pre - image and image are collinear with the center of dilation.

Step2: Analyze each option

• Option 1:

If the pre - image is closer to the center of dilation than the image, since the distance from the center of dilation to the image is \(k\) times the distance from the center of dilation to the pre - image (\(d_{image}=k\times d_{pre - image}\)), when \(k > 1\), \(d_{image}>d_{pre - image}\). So, the pre - image is closer to the point of dilation than the image. This is a correct statement for enlargement.

• Option 2:

If the image is closer to the center of dilation than the pre - image, then \(k=\frac{d_{image}}{d_{pre - image}}<1\) (reduction). So, this is an incorrect statement for enlargement.

• Option 3:

If the scale factor \(k = \frac{1}{6}\in(0,1)\), it is a reduction. So, this is an incorrect statement for enlargement.

• Option 4:

If the scale factor \(k = 6>1\), by the definition of dilation (\(A'B'=k\times AB\) for a line segment \(AB\) in the pre - image and \(A'B'\) in the image), it is an enlargement. This is a correct statement for enlargement.

• Option 5:

The collinearity of vertices of pre - image and image with the center of dilation is a property of dilation (both for reduction and enlargement). So, it does not specifically justify enlargement.

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 \(1\).