Question

A line segment with endpoints (−3,4) and (5,−6) is used to perform a dilation with center (0,0) . After the dilation, the new line segment has a length that is half as long as the original line segment. The dilation has a scale factor of k. Annabelle claims that k must have been 12 . Which statement BEST describes Annabelle's claim? Responses Annabelle's claim is incorrect since the scale factor of dilation was 2 , not 12 . Annabelle's claim is incorrect since the scale factor of dilation was 2 , not 1 2 . Annabelle's claim is incorrect since the scale factor of dilation could have also have been −12 . Annabelle's claim is incorrect since the scale factor of dilation could have also have been - 1 2 . Annabelle's claim is correct since a scale factor of 12 results in a line segment half as long as the original. Annabelle's claim is correct since a scale factor of 1 2 results in a line segment half as long as the original. Annabelle's claim is correct since a dilation of a line segment half as long as the original must have a scale factor of 12 . Annabelle's claim is correct since a dilation of a line segment half as long as the original must have a scale factor of 1 2 .

Answer 1

The best statement to describe Annabelle's claim is: C. Annabelle's claim is correct since a scale factor of 1/2 results in a line segment half as long as the original.

Here's a breakdown of why this is the case:

Scale factor in dilations determines how much the image is enlarged or shrunk compared to the original figure. A scale factor of 1 would mean no change in size, a scale factor greater than 1 would enlarge the image, and a scale factor between 0 and 1 would shrink the image.

In this problem, the new line segment is half as long as the original, meaning it has been shrunk.

The only scale factor among the options that would result in a shrunken image is 1/2. A scale factor of 2 would double the size, and the other options are not relevant to shrinking the image.

Therefore, Annabelle's claim that the scale factor must have been 1/2 is correct.

Scale factor is a multiplier that affects the size of the image.

Scale factors less than 1 shrink the image, while those greater than 1 enlarge it.

A scale factor of 1/2 specifically reduces the size by half.

Question:

Answer 2

Annabelle's claim is incorrect since the scale factor of the dilation was 1/2, not 12.

The scale factor of a dilation is determined by comparing the lengths of corresponding sides of the original figure and the dilated figure.

In this case, Annabelle claimed that the scale factor must be 12 because the new line segment is half the length of the original segment. However, this claim is incorrect.

To find the scale factor, we can compare the lengths of the original segment (10 units) and the dilated segment (5 units).

The scale factor is the ratio of the lengths: 5/10 = 1/2.

Therefore, Annabelle's claim is incorrect since the scale factor of the dilation was 1/2, not 12.