Line Segment AB has endpoints A (2,3) and B (4,1). Determine the scale factor used in the dilation of line segment A' B', centered at the origin.
To determine the scale factor used in the dilation of line segment A'B', we need to calculate the ratio of the lengths of the original line segment AB and the dilated line segment A'B'.
The length of a line segment AB can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, the coordinates of A and B are (2, 3) and (4, 1) respectively. Plugging these values into the distance formula:
d_AB = sqrt((4 - 2)^2 + (1 - 3)^2)
= sqrt(2^2 + (-2)^2)
= sqrt(4 + 4)
= sqrt(8)
= 2sqrt(2)
Now, let's calculate the length of the dilated line segment A'B'. Since the dilation is centered at the origin, the coordinates of A' and B' will be the same as those of A and B, but scaled by the scale factor.
The scale factor is the ratio of the length of the dilated line segment A'B' to the length of the original line segment AB.
So, to calculate the scale factor, we have:
Scale factor = length of A'B' / length of AB
Now, we know that the length of AB is 2sqrt(2), and since the dilation is centered at the origin, the coordinates of A' and B' are (2, 3) and (4, 1) scaled by the scale factor respectively.
By comparing the coordinates, we can see that the scale factor is 1, because the coordinates of A' are (2, 3), which is the same as the original coordinates of A.
Therefore, the scale factor used in the dilation of line segment A'B', centered at the origin, is 1.