Given a dilation with the origin O (0, 0), by observation determine the scale factor "k."

DO,K = (6, 12) (3, 6)

What is the scale factor K = a0

Is the dilation an expansion or contraction? a1

yes

To determine the scale factor "k" of a dilation given two corresponding points, you can use the formula:

k = distance between corresponding points in the image / distance between corresponding points in the pre-image.

In this case, we have two corresponding points: O (0, 0) and D (6, 12) in the pre-image, and O (0, 0) and K (3, 6) in the image.

First, let's find the distance between O and D in the pre-image:
distance_OD = sqrt((6 - 0)^2 + (12 - 0)^2) = sqrt(36 + 144) = sqrt(180)

Next, let's find the distance between O and K in the image:
distance_OK = sqrt((3 - 0)^2 + (6 - 0)^2) = sqrt(9 + 36) = sqrt(45)

Now, we can calculate the scale factor "k" using the formula mentioned earlier:
k = distance_OK / distance_OD = sqrt(45) / sqrt(180)

To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by sqrt(4):
k = (sqrt(45) / sqrt(180)) * (sqrt(4) / sqrt(4))
= (sqrt(45) * sqrt(4)) / (sqrt(180) * sqrt(4))
= sqrt(45 * 4) / sqrt(180 * 4)
= sqrt(180) / sqrt(720)
= sqrt(180) / sqrt(36 * 20)
= sqrt(180) / (6 * sqrt(20))
= (6 * sqrt(5)) / (6 * sqrt(20))
= sqrt(5) / sqrt(20)
= sqrt(1/4)
= 1/2

Therefore, the scale factor k = 1/2.

Since the scale factor k = 1/2, which is less than 1, the dilation is a contraction.