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Question

A coordinate plane's axes range from negative 3 to 7 by 1-unit increments. 3 points are plotted and joined by solid lines. The points are labeled upper A, upper B, and upper C.

AC¯¯¯¯¯¯¯¯ (5,-1) (5,-1)
is a dilation of AB¯¯¯¯¯¯¯¯ (2,-1) (2,-1)
, with point B (2,2)
as the center of dilation. Determine the scale factor.

(1 point)
The scale factor is
.
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Answer 1

To find the scale factor of the dilation from segment AB to segment AC with respect to point B, follow these steps:

1. Identify the coordinates of the points:

   • \( A (2, -1) \)
   • \( B (2, 2) \)
   • \( C (5, -1) \)

2. Calculate the distance from point B to point A (denoted as AB):

   • The distance \( d_{AB} \) can be calculated using the distance formula or by observing the coordinates directly, since points B and A share the same x-coordinate: \[ d_{AB} = |y_B - y_A| = |2 - (-1)| = |2 + 1| = |3| = 3 \]

3. Calculate the distance from point B to point C (denoted as BC):

   • Similarly, calculate \( d_{BC} \): \[ d_{BC} = |y_B - y_C| = |2 - (-1)| = |2 + 1| = |3| = 3 \]
   • Notice that the x-coordinates are different again so we check their squared difference: \[ d_{BC} = \sqrt{(5 - 2)^2 + (-1 - 2)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

4. Calculate the scale factor \( k \): \[ k = \frac{d_{AC}}{d_{AB}} \] However, we will also adjust for the following points. The distance from point B to point A is 3, and from point B to point C needs to be calculated as \( d_{BC} \).

5. The distances lead to the following: \[ k = \frac{3\sqrt{2}}{3} = \sqrt{2} \]

Finally, conclude that the scale factor is:

The scale factor is \( \sqrt{2} \).