Dilate \( \triangle A B C \) using point \( C \) as the center of dilation and a scale factor of 3 to form \( \triangle A^{\prime} B^{\prime} C^{\prime} \). Explain how you determined the coordinates of the dilated figure.

To dilate \( \triangle ABC \) using point \( C \) and a scale factor of 3, you need to multiply the distance from \( C \) to points \( A \) and \( B \) by 3. If \( C \) has coordinates \( (x_C, y_C) \), then the coordinates of the dilated points can be found using the formula: \[ A' = (1 - k)(x_C) + k(x_A), \quad B' = (1 - k)(x_C) + k(x_B) \] where \( k \) is the scale factor (in this case, 3) and \( (x_A, y_A) \) are the original coordinates of points \( A \) and \( B \). Essentially, you can visualize this as moving both points \( A \) and \( B \) away from point \( C \) by a factor of 3. For instance, if point \( A \) is at \( (2, 3) \) and \( C \) is at \( (1, 1) \), you can apply the formula for \( A' \): \[ A' = (1 - 3)(1) + 3(2) = -2 + 6 = 4 \text{ for the x-coordinate} \] \[ A' = (1 - 3)(1) + 3(3) = -2 + 9 = 7 \text{ for the y-coordinate} \] Thus, \( A' \) becomes \( (4, 7) \). Repeat this for point \( B \) to get \( B' \), resulting in the dilated \( \triangle A' B' C' \).