Question

Determine the scale factor used in the dilation of line AB, (9,9) (6,6) with a center point of dilation at the origin.

Answer 1

To determine the scale factor used in the dilation of line AB, we need to compare the lengths of the original line AB and the dilated line.

The length of a line can be calculated using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

For the original line AB, (9,9) and (6,6), the length is:
d = sqrt((6 - 9)^2 + (6 - 9)^2) = sqrt((-3)^2 + (-3)^2) = sqrt(18) = 3sqrt(2)

For the dilated line, we need to find the coordinates of the corresponding points after dilation. The given center of dilation is the origin (0,0).

Point A after dilation:
Ax = 9 * scale factor = 9 * k
Ay = 9 * scale factor = 9 * k

Point B after dilation:
Bx = 6 * scale factor = 6 * k
By = 6 * scale factor = 6 * k

Substituting the coordinates into the distance formula for the dilated line AB:
d = sqrt((Bx - Ax)^2 + (By - Ay)^2)
= sqrt((6k - 9k)^2 + (6k - 9k)^2)
= sqrt((-3k)^2 + (-3k)^2)
= sqrt(18k^2)
= 3k sqrt(2)

Comparing the length of the original line AB (3sqrt(2)) with the dilated line (3k sqrt(2)), we can see that the scale factor used in the dilation is k.

So, the scale factor used in the dilation of line AB is k.