it's geometry

First you need to find the distance of both points from the origin - hence the distance formula (or Pythagorean Theorem, given the second point is the origin (0,0))

The distance from (1,-3) to the origin is given by:

d

_{1}=√[1^{2}+(-3)^{2}] =√[1+9]

=√10

The distance from (4,-12) to the origin is given by:

d

_{2}=√[4^{2}+(-12)^{2}] =√[4

^{2}+4^{2}(-3)^{2}] factoring (-12)^{2}=[4(-3)]^{2}=4^{2}(-3)^{2} =√[4

^{2}(1+9)] factoring the common 4^{2}from both terms =√4

^{2 }√10 =4√10

Hence the dilation factor of the final point (4,-12) with respect to the initial point (1,-3) is given by:

d

_{2}/d_{1}= (4√10) /√10 =4Note: we can see by inspection, that each coordinate is four times as large; i.e. the x-coordinate went from 1 → 4, and the y-coordinate went from -3 → -12