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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

2
2

4
4

14

1 answer

To determine the scale factor used in the dilation of line segment \(AB\) to \(A'B'\) centered at the origin, we can compare the coordinates of points \(A\) and \(B\) with their corresponding dilated points \(A'\) and \(B'\).

  1. Coordinates of points:

    • \(A(1, -3)\)
    • \(B(2, -4)\)
    • \(A'(4, -12)\)
    • \(B'(8, -16)\)

  2. Finding the dilation for point A:

    • The original coordinates of A are \((1, -3)\).
    • The dilated coordinates of \(A'\) are \((4, -12)\).
    • The scale factor \(k\) can be found using the formula: \[ k = \frac{\text{new coordinate}}{\text{original coordinate}} \]
    • For the x-coordinate: \[ k = \frac{4}{1} = 4 \]
    • For the y-coordinate: \[ k = \frac{-12}{-3} = 4 \]

  3. Finding the dilation for point B:

    • The original coordinates of B are \((2, -4)\).
    • The dilated coordinates of \(B'\) are \((8, -16)\).
    • Again, using the scale factor formula:
    • For the x-coordinate: \[ k = \frac{8}{2} = 4 \]
    • For the y-coordinate: \[ k = \frac{-16}{-4} = 4 \]

  4. Final answer: Since both points A and B show the same scale factor of 4 when dilated to \(A'\) and \(B'\), we conclude that the scale factor used in the dilation of line segment \(AB\), centered at the origin, is:

\[ \boxed{4} \]