How do you determine the scale factor of dilation?

If a dilation) (or scaling) is given, it is assumed that its center and a factor are given, so we can construct an image of any point.

Step-by-step explanation:

If center of dilation is point  O  and factor is  f ≠ 0 , any given point  A  is transformed by a dilation into point  A '  such that  (a) points  O ,  A  and  A '  are on the same line; (b) if  f > 0 , points  A  and  A '  are on the same side from center  O ; if  f < 0 , point  O  is in between  A  and  A ' ; (c) Lengths of segments  O A '  and  O A  relate to each other at factor  | f| f| , that is  | O A ' | | O A | = | f |

If these two parameters, the center and the factor, are not known, something must be given to determine them.

If a center  O  is given, to determine a factor we need points  A  and its image (the result of scaling)  A ' .

Then, knowing mutual position of points  O ,  A  and  A '  and lengths of segments  O A  and  O A ' , we can determine a factor, its sign based on position of given points and its absolute value as the ratio of the lengths of segments  O A '  and  O A .

If even a center is not given, we need two pairs of points: sources  

A  and  B  and their images  A ' and  B ' .

Then we determine the center as an intersection of lines  A A '  and  B B '  and then follow the procedure above since a center is known now.

18

step-by-step explanation:

draw the shape and you will see that f and k are actually at adjacent vertices (points of the hexagon that are neighboring).   so find the distance between (1,2) and (4,2).   that is easy since (1,2) and (4,2) is just a vertical distance to count, which is 3.   it is a regular polygon which means all sides are congruent.   you have 6 sides (hexa) so multiply 6 by 3 to receive 18.