A quadrilateral has the following vertices (4, 4) (6, 6) (8, 4) (6, 2) After dilation, the new
quadrilateral has the following vertices (3, 3) (4.5, 4.5), (6, 3) (4.5, 1.5) What scale factor was used to dilate the quadrilateral? (1 point)
To find the scale factor used to dilate the quadrilateral, we can compare the corresponding side lengths of the original quadrilateral and the dilated quadrilateral.
Original quadrilateral side lengths:
1. (4, 4) to (6, 6) = sqrt[(6-4)^2 + (6-4)^2] = sqrt(8) = 2sqrt(2)
2. (6, 6) to (8, 4) = sqrt[(8-6)^2 + (4-6)^2] = sqrt(8) = 2sqrt(2)
3. (8, 4) to (6, 2) = sqrt[(6-8)^2 + (2-4)^2] = sqrt(8) = 2sqrt(2)
4. (6, 2) to (4, 4) = sqrt[(4-6)^2 + (4-2)^2] = sqrt(8) = 2sqrt(2)
Dilated quadrilateral side lengths:
1. (3, 3) to (4.5, 4.5) = sqrt[(4.5-3)^2 + (4.5-3)^2] = sqrt(4.5)
2. (4.5, 4.5) to (6, 3) = sqrt[(6-4.5)^2 + (3-4.5)^2] = sqrt(4.5)
3. (6, 3) to (4.5, 1.5) = sqrt[(4.5-6)^2 + (1.5-3)^2] = sqrt(4.5)
4. (4.5, 1.5) to (3, 3) = sqrt[(3-4.5)^2 + (3-1.5)^2] = sqrt(4.5)
Now, we will compare the original side lengths to the dilated side lengths:
Original side length / Dilated side length
= 2sqrt(2) / sqrt(4.5)
= 2sqrt(2) / (2*sqrt(1.5))
= sqrt(2) / sqrt(1.5)
= sqrt(2 / 1.5)
= sqrt(4/3)
= 2 / sqrt(3)
= 2sqrt(3) / 3
Therefore, the scale factor used to dilate the quadrilateral is 2sqrt(3) / 3.