A pre-image segment with endpoints (3,-2) and (4,2) is dilated to the image segment with endpoints (9,-6) and (12,6). What is the scale factor for the dilation?

Note that each value is multiplied by 3.

To find the scale factor of a dilation, we can compare the lengths of corresponding sides of the pre-image and the image.

First, let's find the length of the pre-image segment. We can use the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

where (x₁, y₁) and (x₂, y₂) are the coordinates of the endpoints. Substitute the coordinates (3, -2) and (4, 2) into the formula:

d_pre = √((4 - 3)² + (2 - (-2))²) = √(1² + 4²) = √(1 + 16) = √17

Next, let's find the length of the image segment using the same formula:

d_image = √((12 - 9)² + (6 - (-6))²) = √(3² + 12²) = √(9 + 144) = √153

Now, we can find the scale factor by dividing the length of the image segment by the length of the pre-image segment:

scale factor = d_image / d_pre = √153 / √17

To simplify the scale factor, we can rationalize the denominator by multiplying both the numerator and denominator by √17:

scale factor = (√153 * √17) / (√17 * √17) = √(153 * 17) / √(17 * 17) = √2601 / 17 = 51 / 17

Therefore, the scale factor for the dilation is 51/17.