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Find the scale factor of the dilation, with respect to the origin, for the mapping of (1, -3) ? (4, -12).

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1 Answer

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:
 
d1=√[12+(-3)2]
  =√[1+9]
  =√10
 
The distance from (4,-12) to the origin is given by:
 
d2=√[42+(-12)2]
  =√[42+42(-3)2]  factoring (-12)2=[4(-3)]2=42(-3)2
  =√[42(1+9)]  factoring the common 42 from both terms
  =√42 √10  
  =4√10  
 
Hence the dilation factor of the final point (4,-12) with respect to the initial point (1,-3) is given by:
 
d2/d1 = (4√10) /√10 =4
 
Note: 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
 

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