Question

justifying the coordinates of a pre - image
the segments shown are dilations of each other about the origin. which statement could be true?
the coordinate (1, 0) is from a dilation using the scale factor of \\(\frac{1}{5}\\).
the coordinate (5, 0) is from a dilation using the scale factor of 4.
the coordinate (1, 0) is from a dilation using the scale factor of 5.
the coordinate (5, 0) is from a dilation using the scale factor of \\(\frac{1}{5}\\).

Explanation:

Step1: Recall dilation formula

For dilation about origin: $(x', y') = k(x, y)$, where $k$ = scale factor, $(x,y)$ = pre-image, $(x',y')$ = image.

Step2: Test first option

Check if $(1,0) = \frac{1}{5}(5,0)$:
$\frac{1}{5} \times 5 = 1$, $\frac{1}{5} \times 0 = 0$. This holds true.

Step3: Test second option

Check if $(5,0) = 4(1,0)$:
$4 \times 1 = 4
eq 5$. This is false.

Step4: Test third option

Check if $(1,0) = 5(5,0)$:
$5 \times 5 = 25
eq 1$. This is false.

Step5: Test fourth option

Check if $(5,0) = \frac{1}{5}(1,0)$:
$\frac{1}{5} \times 1 = \frac{1}{5}
eq 5$. This is false.

Answer:

The coordinate (1, 0) is from a dilation using the scale factor of $\frac{1}{5}$.