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

2
2

−4
negative 4

4
4

Answer 1

To determine the scale factor used in the dilation of line segment \(AB\) (from points \(A(1, -3)\) and \(B(2, -4)\)) to points \(A'(4, -12)\) and \(B'(8, -16)\), we can use the formula for dilation centered at the origin:

\[
k = \frac{(x', y')}{(x, y)}
\]

where \(k\) is the scale factor, \((x, y)\) are the original coordinates, and \((x', y')\) are the coordinates after dilation.

Let's calculate the scale factor for point \(A\):

1. For point \(A(1, -3)\) and \(A'(4, -12)\):
- The coordinates of \(A\) are \((1, -3)\) and of \(A'\) are \((4, -12)\).

For the x-coordinates:
\[
k_x = \frac{4}{1} = 4
\]

For the y-coordinates:
\[
k_y = \frac{-12}{-3} = 4
\]

Since both calculations give \(k = 4\), the scale factor for point \(A\) is \(4\).

Next, let's check point \(B\):

2. For point \(B(2, -4)\) and \(B'(8, -16)\):
- The coordinates of \(B\) are \((2, -4)\) and the coordinates of \(B'\) are \((8, -16)\).

For the x-coordinates:
\[
k_x = \frac{8}{2} = 4
\]

For the y-coordinates:
\[
k_y = \frac{-16}{-4} = 4
\]

Again, both calculations give \(k = 4\).

Thus, the scale factor used in the dilation of line segment \(AB\) centered at the origin is:

\[
\boxed{4}
\]