A coordinate plane with four quadrants shows x-axis ranging from negative 5 to 12 in increments of 1 and the y-axis from 17 to negative 4 in increments of 1. Points upper A, upper B, upper A prime, and upper B prime are plotted on quadrant 1. The plotted points are as follows: Upper A is left parenthesis 2 comma 3 right parenthesis and upper B is left parenthesis 4 comma 1 right parenthesis. The points are joined by a line. Upper A prime is plotted as left parenthesis 6 comma 9 right parenthesis and upper B prime as left parenthesis 12 comma 3 right parenthesis. The points are joined by a line.

Question

determine the scale factor used in the dilation of line segment AB, centered at the origin.

the scale factor is __

Answer 1

To find the scale factor used in the dilation of line segment \( AB \) (from points \( A(2, 3) \) and \( B(4, 1) \)) to line segment \( A'B' \) (with \( A'(6, 9) \) and \( B'(12, 3) \)), we can use the formula for scale factor in dilation, which relates the coordinates of the original points and the dilated points.

The scale factor \( k \) can be determined by comparing the coordinates of points A and A' (or B and B').

Steps:

1. Find the scale factor for point A to A':

   • Original coordinates of A: \( (2, 3) \)
   • Dilated coordinates of A': \( (6, 9) \)

   For the x-coordinate: \[ k_x = \frac{x'}{x} = \frac{6}{2} = 3 \]

   For the y-coordinate: \[ k_y = \frac{y'}{y} = \frac{9}{3} = 3 \]

   Thus, the scale factor from A to A' is \( k = 3 \).

2. Now, let's check the scale factor for point B to B':

   • Original coordinates of B: \( (4, 1) \)
   • Dilated coordinates of B': \( (12, 3) \)

   For the x-coordinate: \[ k_x = \frac{x'}{x} = \frac{12}{4} = 3 \]

   For the y-coordinate: \[ k_y = \frac{y'}{y} = \frac{3}{1} = 3 \]

   The scale factor from B to B' is also \( k = 3 \).

Conclusion:

Since both calculations yield the same scale factor, the scale factor used in the dilation of line segment \( AB \), centered at the origin, is

\[ \boxed{3} \]