Determine the scale factor used in the dilation of line AB , with a center point of dilation at the origin. A (-3, 4), B (2,-4), A' (-6,8), B' (4, -8)
The scale factor can be determined by finding the ratio of the lengths of the corresponding sides of the dilated and original figures.
The length of the original line AB can be found using the distance formula, which is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
d(A,B) = √((2 - (-3))^2 + (-4 - 4)^2)
= √(5^2 + (-8)^2)
= √(25 + 64)
= √89
The length of the dilated line A'B' can also be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
d(A',B') = √((4 - (-6))^2 + (-8 - 8)^2)
= √(10^2 + (-16)^2)
= √(100 + 256)
= √356
The scale factor can now be determined by dividing the length of the dilated line by the length of the original line:
Scale Factor = d(A',B') / d(A,B)
= √356 / √89
This can be simplified by rationalizing the denominator:
Scale Factor = (√356 / (√89) ) * (√89 / √89 )
= √( 356 * 89 ) / 89
The scale factor is √( 356 * 89 ) / 89.
Note: Depending on the level of accuracy desired, this value can be further simplified or evaluated numerically.