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

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

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To dilate triangle $$ \triangle ABC $$ using point $$ C $$ as the center of dilation with a scale factor of 3 to form $$ \triangle A'B'C' $$, follow these steps:

1. **Identify the coordinates of triangle $$ \triangle ABC $$**:
   Let's assume the coordinates of $$ A $$, $$ B $$, and $$ C $$ are $$ A(x_1, y_1) $$, $$ B(x_2, y_2) $$, and $$ C(x_3, y_3) $$, respectively.

2. **Determine the coordinates of the center of dilation $$ C $$**:
   Since $$ C $$ is the center of dilation, its coordinates are $$ C(x_3, y_3) $$.

3. **Determine the scale factor**:
   The scale factor is given as 3.

4. **Dilate the triangle**:
   To dilate a point $$ (x, y) $$ with respect to the center $$ (x_0, y_0) $$ and a scale factor $$ k $$, you use the following formulas:
   $$
   x' = k(x - x_0) + x_0
   $$
   $$
   y' = k(y - y_0) + y_0
   $$
   where $$ (x', y') $$ are the coordinates of the dilated point.

Apply these formulas to each of the vertices of $$ \triangle ABC $$ to find the coordinates of $$ \triangle A'B'C' $$.

- For point $$ A $$:
     $$
     A'(x_1', y_1') = 3(A(x_1, y_1) - C(x_3, y_3)) + C(x_3, y_3)
     $$
     $$
     A'(x_1', y_1') = 3((x_1 - x_3), (y_1 - y_3)) + (x_3, y_3)
     $$
     $$
     A'(x_1', y_1') = (3x_1 - 3x_3 + x_3, 3y_1 - 3y_3 + y_3)
     $$
     $$
     A'(x_1', y_1') = (3x_1 - 2x_3, 3y_1 - 2y_3)
     $$

- For point $$ B $$:
     $$
     B'(x_2', y_2') = 3(B(x_2, y_2) - C(x_3, y_3)) + C(x_3, y_3)
     $$
     $$
     B'(x_2', y_2') = 3((x_2 - x_3), (y_2 - y_3)) + (x_3, y_3)
     $$
     $$
     B'(x_2', y_2') = (3x_2 - 3x_3 + x_3, 3y_2 - 3y_3 + y_3)
     $$
     $$
     B'(x_2', y_2') = (3x_2 - 2x_3, 3y_2 - 2y_3)
     $$

- For point $$ C $$:
     Since $$ C $$ is the center of dilation, the coordinates of $$ C' $$ are the same as the coordinates of $$ C $$:
     $$
     C'(x_3, y_3)
     $$

5. **Summarize the coordinates of the dilated triangle $$ \triangle A'B'C' $$**:
   $$
   A'(x_1', y_1') = (3x_1 - 2x_3, 3y_1 - 2y_3)
   $$
   $$
   B'(x_2', y_2') = (3x_2 - 2x_3, 3y_2 - 2y_3)
   $$
   $$
   C'(x_3, y_3)
   $$

These are the coordinates of the dilated triangle $$ \triangle A'B'C' $$.
To find the coordinates of the dilated triangle $$ \triangle A'B'C' $$, use the formula $$ (kx + x_0, ky + y_0) $$ with $$ k = 3 $$ and $$ C(x_3, y_3) $$ as the center. The coordinates of $$ A' $$, $$ B' $$, and $$ C' $$ are then $$ (3x_1 - 2x_3, 3y_1 - 2y_3) $$, $$ (3x_2 - 2x_3, 3y_2 - 2y_3) $$, and $$ (x_3, y_3) $$, respectively.

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.

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