Line Segment AB has endpoints A (2,3) and B (4,1). Determine the scale factor used in the dilation of line segment AB, centered at the origin.

Answer 1

To determine the scale factor of a dilation, we need to find the ratio of the lengths of the line segments before and after the dilation. In this case, the line segment AB has endpoints A(2,3) and B(4,1), so its length is given by the distance formula:

AB = sqrt((4 - 2)^2 + (1 - 3)^2)
= sqrt(2^2 + (-2)^2)
= sqrt(4 + 4)
= sqrt(8)
= 2*sqrt(2)

Now, to find the length of the dilated line segment, we multiply the length before dilation by the scale factor. Since the dilation is centered at the origin, the scale factor is the same in both the x and y directions. Therefore, the scale factor is the same as the ratio of the x coordinates or the y coordinates. In this case, we can calculate the ratio of the x coordinates:

scale factor = (4 - 0) / (2 - 0)
= 4/2
= 2

The scale factor used in the dilation of line segment AB, centered at the origin, is 2.