Determine the scale factor used in the dilation of line segment AB, centered at the origin.
A(4,-12), B(8,-16)
a)-4
b)1/4
c)4
d)2
To find the scale factor used in the dilation, we need to calculate the ratio of the lengths of the original line segment AB to the image of line segment AB.
The length of AB can be found using the distance formula:
AB = sqrt((8-4)^2 + (-16--12)^2)
= sqrt(4^2 + 4^2)
= sqrt(32)
= 4*sqrt(2)
The length of the image of AB can be found the same way, but using the coordinates of the image points:
AB' = sqrt((2-0)^2 + (-4-0)^2)
= sqrt(2^2 + 4^2)
= sqrt(20)
= 2*sqrt(5)
The scale factor is the ratio of the lengths:
scale factor = AB'/AB
= (2*sqrt(5)) / (4*sqrt(2))
= sqrt(5) / (2*sqrt(2))
= sqrt(5/2) / sqrt(2)
= sqrt(5/4)
= sqrt(1.25)
= 1.11803398875
Therefore, the scale factor used in the dilation of line segment AB is approximately 1.118.
None of the given options match this value exactly.