Use the image to answer the question.
A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted. A coordinate plane with four quadrants shows an x axis capped with arrows and ranging from negative 11 to 10 in 1 unit increments and a y axis capped with arrows and ranging from 12 to negative 12 in 1 unit increments. Point upper A is plotted as left parenthesis negative 3 comma 4 right parenthesis and point upper B is plotted as left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends. Point upper A prime is plotted as left parenthesis negative 6 comma 8 right parenthesis and point upper B prime is plotted as left parenthesis 4 comma negative 8 right parenthesis. The points are joined by a line that extends beyond the points and is capped with arrows at both ends.
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, we can compare the lengths of corresponding line segments before and after the dilation.
The length of line segment AB is the distance between points A(-3, 4) and B(2, -4). Using the distance formula, we can calculate:
d_AB = √[(2 - (-3))^2 + (-4 - 4)^2]
= √[(5)^2 + (-8)^2]
= √[25 + 64]
= √89
The length of line segment A'B' is the distance between points A'(-6, 8) and B'(4, -8). Using the distance formula again, we can calculate:
d_A'B' = √[(4 - (-6))^2 + (-8 - 8)^2]
= √[(10)^2 + (-16)^2]
= √[100 + 256]
= √356
The scale factor can be found by dividing the length of the dilated line segment by the length of the original line segment:
scale factor = d_A'B' / d_AB
= √356 / √89
Since the square root cannot be simplified further, the scale factor is √356 / √89.