Transformations and Similarity Unit Test

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Question
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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

−4
negative 4

4
4

2
2

1/4

Answer 1

To determine the scale factor used in the dilation of line segment AB, we can compare the lengths of AB and A' B'.

The length of AB can be calculated using the distance formula:
AB = √((x2 - x1)^2 + (y2 - y1)^2)
AB = √((2 - 1)^2 + (-4 - (-3))^2)
AB = √((1)^2 + (-1)^2)
AB = √(1 + 1)
AB = √2

The length of A' B' can also be calculated using the distance formula:
A' B' = √((x2' - x1')^2 + (y2' - y1')^2)
A' B' = √((8 - 4)^2 + (-16 - (-12))^2)
A' B' = √((4)^2 + (-4)^2)
A' B' = √(16 + 16)
A' B' = √32
A' B' = 4√2

The scale factor can be found by dividing the length of A' B' by the length of AB:
Scale factor = A' B' / AB
Scale factor = 4√2 / √2
Scale factor = 4

Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.