To find the coordinates of the points A', B', and C' after a dilation with a scale factor of 3 from the origin, we can use the formula for dilation:
\[ (x', y') = (kx, ky) \]
where \((x, y)\) are the original coordinates of the point, \(k\) is the scale factor, and \((x', y')\) are the coordinates after dilation.
Given the scale factor \(k = 3\) and the center of dilation at the origin \((0, 0)\), we will apply the formula to each vertex of triangle ABC.
- 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 the dilation are:
- \(A' = (-6, -6)\)
- \(B' = (-12, -9)\)
- \(C' = (-6, -15)\)
Therefore, the coordinates of the points A', B', and C' after the dilation are:
\[ A'(-6, -6), \quad B'(-12, -9), \quad C'(-6, -15) \]