To determine the scale factor used in the dilation of line segment AB, let's first identify the original coordinates of the points A and B and their dilated coordinates.
The original points are:
- A (9,9)
- B (6,6)
The dilated points are:
- A' (3,3)
- B' (2,2)
Next, we can find the scale factor \( k \) by using the formula for dilation, which is given by:
\[ A' = k \cdot A \] \[ B' = k \cdot B \]
We can get the scale factor from the coordinates of either point A or point B.
For point A:
Using the coordinates of A:
- Original A: (9, 9)
- Dilated A': (3, 3)
We can find \( k \): \[ 3 = k \cdot 9 \]
To find \( k \), we divide both sides by 9: \[ k = \frac{3}{9} = \frac{1}{3} \]
Now checking for B:
- Original B: (6, 6)
- Dilated B': (2, 2)
Using B: \[ 2 = k \cdot 6 \]
To find \( k \), we divide both sides by 6: \[ k = \frac{2}{6} = \frac{1}{3} \]
Thus, the scale factor for the dilation is \( \frac{1}{3} \).
The answer is \( \frac{1}{3} \).
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