Answer: The correct option is (B) \(\dfrac{2}{3}.\)
Step-by-step explanation: Given that the co-ordinates of the end-points of a line segment AB are A(2, 9) and B(5, 8). After being dilated about the origin (0, 0), the co-ordinates of the end-points of image A'B' are \(A^\prime\left(\dfrac{4}{3},6\right)\) and \(B^\prime\left(\dfrac{10}{3},\dfrac{16}{3}\right).\)
We are to find the scale factor of the dilation.
The scale factor of the dilation will be
\(S=\dfrac{\textup{length of the image line}}{\textup{length of the original line}}.\)
The lengths of the lines AB and A'B' are calculated using distance formula as follows:
\(AB=\sqrt{(5-2)^2+(8-9)^2}=\sqrt{9+1}=\sqrt{10},\\\\\\A'B'=\sqrt{\left(\dfrac{10}{3}-\dfrac{4}{3}\right)^2+\left(\dfrac{16}{3}-6\right)^2\right)}=\sqrt{4+\dfrac{4}{9}}=\sqrt{\dfrac{40}{9}}=\dfrac{2}{3}\sqrt{10}~\textup{units}.\)
Therefore, the required scale factor of dilation is
\(S=\dfrac{A'B'}{AB}=\dfrac{\frac{2}{3}\sqrt{10}}{\sqrt{10}}=\dfrac{2}{3}.\)
Thus, the scale factor of the dilation is \(\dfrac{2}{3}.\)
Option (B) is CORRECT.
