Use the image to answer the question.
A coordinate plane with four quadrants showsthe x-axis ranging from negative 4 to 12 in increments of 1 and the y-axis from 1 to negative 16 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 4. The plotted points are as follows: Upper A is left parenthesis 1 comma negative 3 right parenthesis and upper B is left parenthesis 2 comma negative 4 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 4 comma negative 12 right parenthesis and upper B prime as left parenthesis 8 comma negative 16 right parenthesis. The points are joined by a line.
Determine the scale factor used in the dilation of line segment AB , centered at the origin.
(1 point)
Responses
14
Start Fraction 1 over 4 End Fraction
−4
negative 4
2
4
To determine the scale factor used in the dilation of line segment AB, we need to find the ratio of the lengths of AB' to AB.
The length of AB is the distance between the points (1, -3) and (2, -4), which can be found using the distance formula:
AB = sqrt((2-1)^2 + (-4-(-3))^2) = sqrt(1^2 + (-1)^2) = sqrt(2)
The length of AB' is the distance between the points (4, -12) and (8, -16):
AB' = sqrt((8-4)^2 + (-16-(-12))^2) = sqrt(4^2 + (-4)^2) = sqrt(16 + 16) = sqrt(32) = 4sqrt(2)
The scale factor is the ratio of AB' to AB:
Scale factor = AB'/AB = (4sqrt(2))/sqrt(2) = 4
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.