line segment lm is dilated to create lm using point q as the center of dilation and a scale factor of 2…

Explanation:

Step1: Recall dilation property

In a dilation with center (Q) and scale - factor (k = 2), if we consider a point (M) and its image (M'), the ratio of the distance from the center of dilation to the image point and the distance from the center of dilation to the original point is equal to the scale - factor, i.e., (\frac{QM'}{QM}=k).

Step2: Identify given values

We know that (k = 2), and from the figure, we can see that (QM') can be found using the Pythagorean theorem in the right - triangle formed by the segments related to the dilation. However, we can also use the fact that if we assume the distance from (Q) to (M) is (x), and the distance from (Q) to (M') is (y), (y = 2x). Also, from the figure, we can observe that if we consider the right - triangle with legs related to the dilation, we know that the distance from (Q) to (M') is (6) (since the vertical distance from (Q) to (L') is (8) and the vertical distance from (Q) to (L) is (4), and using similar right - triangle relationships). Since (y = 2x) and (y=6) (by observing the figure's geometric relationships), then (x = 3).

Answer:

3