Use the image to answer the question.

Question

A coordinate plane with four quadrants shows an x axis capped with arrows ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows ranging from 15 to negative 4 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted in quadrant 1. Point upper A is plotted as left parenthesis 9 comma 9 right parenthesis and point upper B is plotted as left parenthesis 6 comma 6 right parenthesis. The points are plotted on a line segment. Point upper A prime is plotted as left parenthesis 3 comma 3 right parenthesis and upper B prime is plotted as left parenthesis 2 comma 2 right parenthesis. The points are plotted on a line segment.

Determine the scale factor used in the dilation of line segment AB , with a center point of dilation at the origin.

(1 point)
Responses

−13
negative Start Fraction 1 over 3 End Fraction

13
Start Fraction 1 over 3 End Fraction

−3
negative 3

3
3

Answer 1

To determine the scale factor, we need to compare the lengths of line segment AB and line segment A' B'.

The length of line segment AB can be found using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
AB = √((6 - 9)^2 + (6 - 9)^2)
AB = √((-3)^2 + (-3)^2)
AB = √(9 + 9)
AB = √18

The length of line segment A' B' can be found using the distance formula as well:
A' B' = √((x2 - x1)^2 + (y2 - y1)^2)
A' B' = √((2 - 3)^2 + (2 - 3)^2)
A' B' = √((-1)^2 + (-1)^2)
A' B' = √(1 + 1)
A' B' = √2

The scale factor can be found by dividing the length of AB by the length of A' B':
Scale factor = AB / A' B'
Scale factor = √18 / √2
Scale factor = (√18 / √2) * (√2 / √2)
Scale factor = √36 / √4
Scale factor = 6 / 2
Scale factor = 3

Therefore, the scale factor used in the dilation of line segment AB with a center point of dilation at the origin is 3.