To determine the scale factor of the dilation \( AC \) of line segment \( AB \) with point \( B \) as the center of dilation, we can use the distance formula.
Let's denote:
- Point \( A \) has coordinates \( (x_A, y_A) \)
- Point \( B \) has coordinates \( (x_B, y_B) \)
- Point \( C \) has coordinates \( (x_C, y_C) \)
To find the scale factor \( k \), we use the formula:
\[ k = \frac{d(B, C)}{d(B, A)} \]
where \( d(B, A) \) is the distance from point \( B \) to point \( A \), and \( d(B, C) \) is the distance from point \( B \) to point \( C \).
Given the coordinates:
- \( A = (1, -1) \)
- \( B = (3, -2) \)
- \( C = (6, 5) \)
Now we can calculate the distances:
-
Distance \( d(B, A) \) \[ d(B, A) = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(3 - 1)^2 + (-2 + 1)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
-
Distance \( d(B, C) \) \[ d(B, C) = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(6 - 3)^2 + (5 + 2)^2} = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58} \]
-
Calculate scale factor \( k \) Now compute \( k \): \[ k = \frac{d(B, C)}{d(B, A)} = \frac{\sqrt{58}}{\sqrt{5}} = \sqrt{\frac{58}{5}} \approx \sqrt{11.6} \approx 3.41 \]
So, the scale factor is approximately \( 3.41 \).
However, if you need an exact value, say in a simpler ratio form or a specific decimal, you can provide it as: \[ \frac{\sqrt{58}}{\sqrt{5}} \text{ or } \sqrt{\frac{58}{5}} \] Please let me know if you want a different representation!