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 .

Answer 1

To find the scale factor for a dilation, we need to compare the lengths of the original line segment and the new line segment after the dilation. The correct scale factor for the dilation in this case is 1/2.

Step-by-step explanation:

To find the scale factor for a dilation, we need to compare the lengths of the original line segment and the new line segment after the dilation.

The length of the original line segment can be found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

For the given endpoints (−3,4) and (5,−6), the length of the original line segment is

√((5 - (-3))^2 + (-6 - 4)^2) = √(64 + 100)

= √164.

The length of the new line segment is half of the original length, so we have √164 * k = (√164)/2.

Solving for k, we get k = 1/2.

Based on this calculation, Annabelle's claim that the scale factor is 12 is incorrect.

The correct scale factor for the dilation is 1/2.