To find the coordinates of the points \( A' \), \( B' \), and \( C' \) after a dilation with a scale factor of 3 and the center of dilation at the origin, we can use the formula for dilation. The formula for the coordinates of a point \( (x, y) \) after a dilation from the origin by a scale factor \( k \) is:
\[ (x', y') = (kx, ky) \]
Given the vertices of triangle \( ABC \):
- Point \( A(-2, -2) \)
- Point \( B(-4, -3) \)
- Point \( C(-2, -5) \)
Let's calculate the new coordinates.
-
For point \( A(-2, -2) \): \[ A' = (3 \cdot -2, 3 \cdot -2) = (-6, -6) \]
-
For point \( B(-4, -3) \): \[ B' = (3 \cdot -4, 3 \cdot -3) = (-12, -9) \]
-
For point \( C(-2, -5) \): \[ C' = (3 \cdot -2, 3 \cdot -5) = (-6, -15) \]
Thus, the coordinates after dilation are:
- \( A'(-6, -6) \)
- \( B'(-12, -9) \)
- \( C'(-6, -15) \)
So, the final answer is:
- \( A' = (-6, -6) \)
- \( B' = (-12, -9) \)
- \( C' = (-6, -15) \)