To apply a scale factor of 2.5 and a center of dilation at the origin to each point of the triangle with vertices \( B = (0, 0) \), \( A = (-2, -2) \), and \( C = (2, -4) \), we can use the formula for dilation:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the new coordinates after dilation, and \( k \) is the scale factor.
Given that the scale factor \( k = 2.5 \):
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For point \( A = (-2, -2) \): \[ x' = 2.5 \cdot (-2) = -5 \] \[ y' = 2.5 \cdot (-2) = -5 \] So, the new point \( A' = (-5, -5) \).
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For point \( B = (0, 0) \): \[ x' = 2.5 \cdot 0 = 0 \] \[ y' = 2.5 \cdot 0 = 0 \] So, the new point \( B' = (0, 0) \).
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For point \( C = (2, -4) \): \[ x' = 2.5 \cdot 2 = 5 \] \[ y' = 2.5 \cdot (-4) = -10 \] So, the new point \( C' = (5, -10) \).
Thus, the coordinates of the new triangle after the dilation by a scale factor of 2.5 about the origin are:
- \( A' = (-5, -5) \)
- \( B' = (0, 0) \)
- \( C' = (5, -10) \)