On a grid there are two lines with the following points =
Line 1 : A = (9, 9)
B = (6, 6)
Line 2 : A = (3, 3)
B = (2, 2)
Determine the scale factor used in the dilation of line AB , with a center point of dilation at the origin.
To determine the scale factor used in the dilation of line AB with a center point of dilation at the origin, we need to compare the distances between the points on the original line and the points on the dilated line.
For Line 1:
The distance between points A and B on the original line is √[(9-6)^2 + (9-6)^2] = √[(3^2) + (3^2)] = √(9 + 9) = √18.
The distance between points A and B on the dilated line is √[(3*9 - 3*6)^2 + (3*9 - 3*6)^2] = √[(27 - 18)^2 + (27 - 18)^2] = √[(9^2) + (9^2)] = √(81 + 81) = √162.
Therefore, the scale factor for Line 1 is √162/√18 = √9 = 3.
For Line 2:
The distance between points A and B on the original line is √[(3-2)^2 + (3-2)^2] = √[(1^2) + (1^2)] = √(1 + 1) = √2.
The distance between points A and B on the dilated line is √[(3*3 - 3*2)^2 + (3*3 - 3*2)^2] = √[(9 - 6)^2 + (9 - 6)^2] = √[(3^2) + (3^2)] = √(9 + 9) = √18.
Therefore, the scale factor for Line 2 is √18/√2 = √9 = 3.
In both cases, the scale factor used in the dilation of the line is 3.