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

Question

it's geometry 

Answer 1

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