To find the length of the image of AB after a dilation with a center at the origin and a scale factor of 1/5, you can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
The endpoints of AB are (2, 3) and (2, 5). The dilation with a scale factor of 1/5 means that both the x-coordinate and the y-coordinate of each point will be multiplied by 1/5.
For the first point, (2, 3), after dilation:
x1' = 2 * (1/5) = 2/5
y1' = 3 * (1/5) = 3/5
For the second point, (2, 5), after dilation:
x2' = 2 * (1/5) = 2/5
y2' = 5 * (1/5) = 1
Now, you can calculate the distance between the two transformed points:
Distance = √((x2' - x1')^2 + (y2' - y1')^2)
Distance = √((2/5 - 2/5)^2 + (1 - 3/5)^2)
Distance = √(0^2 + (2/5)^2)
Distance = √(4/25)
Distance = 2/5
So, the length of the image of AB after dilation is 2/5 units, which corresponds to option (b).