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.
To determine the scale factor used in the dilation of line segment AB, we need to compare the lengths of the original line AB with the lengths of the corresponding line A' B' after the dilation.
The length of line segment AB is given by the distance formula:
√((2-1)² + (-4-(-3))²) = √(1² + (-1)²) = √(1+1) = √2
The length of line segment A' B' is given by the distance formula:
√((8-4)² + (-16-(-12))²) = √(4² + (-4)²) = √(16+16) = √32
Therefore, the scale factor is the ratio of the lengths of line segment A' B' to AB:
√32 / √2 = √(32/2) = √16 = 4
The scale factor used in the dilation of line segment AB, centered at the origin, is 4.