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)
To determine the scale factor used in the dilation, you can compare the lengths of the original line AB and the corresponding line segment A'B'.
The length of line AB can be found using the distance formula:
AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
AB = √[(2 - (-3))² + (-4 - 4)²]
AB = √[5² + (-8)²]
AB = √[25 + 64]
AB = √89
The length of line segment A'B' can also be found using the distance formula:
A'B' = √[(x₂ - x₁)² + (y₂ - y₁)²]
A'B' = √[(4 - (-6))² + (-8 - 8)²]
A'B' = √[10² + (-16)²]
A'B' = √[100 + 256]
A'B' = √356
The scale factor is the ratio of the lengths of the corresponding line segments:
Scale factor = Length of A'B' / Length of AB
Scale factor = √356 / √89
To simplify this, we can rationalize the denominator:
Scale factor = (√356 / √89) * (√89 / √89)
Scale factor = √(356 * 89) / √(89 * 89)
Scale factor = √31724 / √7921
Scale factor = √4
Therefore, the scale factor used in the dilation of line AB with a center point of dilation at the origin is 2.