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 $A^{\prime }{B}^{\prime }$ (with $A^{\prime }(6,9)$ and $B^{\prime }(12,3)$), 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: $$k_x=\frac{x^{\prime }}{x}=\frac{6}{2}=3$$

   For the y-coordinate: $$k_y=\frac{y^{\prime }}{y}=\frac{9}{3}=3$$

   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: $$k_x=\frac{x^{\prime }}{x}=\frac{12}{4}=3$$

   For the y-coordinate: $$k_y=\frac{y^{\prime }}{y}=\frac{3}{1}=3$$

   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

$$\boxed{{\displaystyle 3}}$$