In order to solve this problem, we need to find the coordinates of some points after the dilation of 1/3. This requires us to know what is the center of the dilation, i.e., the point in the plane about which the points are expanded or contracted.
Since the center of the dilation wasn't given, we will assume it is the origin (0, 0).
Therefore, the distance of each point from the origin will be rescaled by the factor of 1/3. This means each coordinate will be transformed as:
(x, y) → (x/3, y/3)
A way to see why the coordinates change in this way is the following: let's call d the distance of a point (x, y) from the center:
d = √(x²+y²)
Then the new distance d' after the dilation has to be d/3:
d' = d/3 = √(x²+y²)/3 = √[(x²+y²)/3²] = √[(x/3)²+(y/3)²]
So, to get the new distance d' = d/3, each coordinate must be transformed by the factor 1/3.
Now, for the given points, we have:
A(3, 6) → A'(1, 2)
B(6, 9) → B'(2, 3)
C(6, 12) → C'(2, 4)