Final answer:
The scale factor for this dilation with the center at the origin is 8/9.
Step-by-step explanation:
To find the scale factor for this dilation, we need to compare the distance between the origin (0,0) and the point t (9,6) with the distance between the origin and the preimage point T (12,8). The formula for calculating the scale factor is:
Scale factor = (distance between T and origin) / (distance between t and origin)
Distance between T and origin = √((12-0)^2 + (8-0)^2) = √(144 + 64) = √208
Distance between t and origin = √((9-0)^2 + (6-0)^2) = √(81 + 36) = √117
Therefore, the scale factor is: √208 / √117 = 8√13 / 9√13 = 8/9