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
1 answer
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.