determine the scale factor used in the dilation of line segment AB with a center point of dilation at the origin
A.1/3
B.-3
C.-1/3
D.3
To determine the scale factor used in the dilation of line segment AB with a center point of dilation at the origin, we need to find the ratio of the length of the dilated line segment to the original length of the line segment.
Let's say the coordinates of point A are (x1, y1) and the coordinates of point B are (x2, y2).
The distance between points A and B can be calculated using the distance formula:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
Next, let's dilate the line segment AB by a scale factor of k using the center point O(0, 0).
The coordinates of the dilated points A' and B' will be (kx1, ky1) and (kx2, ky2) respectively.
The distance between the dilated points A' and B' can be calculated as:
A'B' = √[(kx2 - kx1)^2 + (ky2 - ky1)^2]
= √[k^2(x2 - x1)^2 + k^2(y2 - y1)^2]
= k * √[(x2 - x1)^2 + (y2 - y1)^2]
= k * AB
Therefore, the scale factor used in the dilation is k.
Since AB = √[(x2 - x1)^2 + (y2 - y1)^2] and A'B' = k * AB,
The scale factor k = A'B' / AB
So, the correct answer is:
D. 3