To find the new triangle that is dilated from the original triangle with the given vertices \( a = (-2, -2) \), \( b = (0, 0) \), and \( c = (2, -4) \) using a scale factor of \( 2.5 \) and a center of dilation at the origin, we can apply the dilation transformation to each vertex.
The formula for dilation with a center at the origin and a scale factor \( k \) is given by:
\[ (x', y') = (k \cdot x, k \cdot y) \]
where \( (x, y) \) are the coordinates of the original points and \( (x', y') \) are the coordinates of the new points after dilation.
Using a scale factor \( k = 2.5 \):
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Dilation of point \( a = (-2, -2) \): \[ a' = (2.5 \cdot -2, 2.5 \cdot -2) = (-5, -5) \]
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Dilation of point \( b = (0, 0) \): \[ b' = (2.5 \cdot 0, 2.5 \cdot 0) = (0, 0) \]
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Dilation of point \( c = (2, -4) \): \[ c' = (2.5 \cdot 2, 2.5 \cdot -4) = (5, -10) \]
So the coordinates of the new triangle after dilation are:
- \( a' = (-5, -5) \)
- \( b' = (0, 0) \)
- \( c' = (5, -10) \)
Thus, the new triangle vertices are: \[ \text{New triangle: } a' = (-5, -5), b' = (0, 0), c' = (5, -10) \]