To dilate a triangle about a point, you can use the following formula for the coordinates of the dilated point:
\[ (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c)) \]
where \( (x, y) \) are the original coordinates of the point, \( (x_c, y_c) \) is the center of dilation, \( k \) is the scale factor, and \( (x', y') \) are the coordinates after dilation.
Here, the center of dilation is \( (2, 2) \). The coordinates of the original triangle's vertices are:
- \( A(5, 5) \)
- \( B(1, 7) \)
- \( C(3, 2) \)
Before applying the dilation, you need to choose a scale factor \( k \). If you do not specify a scale factor, we'll assume \( k = 1 \) (which would mean no actual dilation) or \( k \) can be any positive number indicating how much you want to dilate.
Assuming we are dilating with a scale factor of \( k = 2 \) as an example, let's calculate the new coordinates of the vertices.
For A(5, 5):
\[ x' = 2 + 2(5 - 2) = 2 + 2 \cdot 3 = 2 + 6 = 8 \] \[ y' = 2 + 2(5 - 2) = 2 + 2 \cdot 3 = 2 + 6 = 8 \]
So, \( A'(8, 8) \).
For B(1, 7):
\[ x' = 2 + 2(1 - 2) = 2 + 2 \cdot (-1) = 2 - 2 = 0 \] \[ y' = 2 + 2(7 - 2) = 2 + 2 \cdot 5 = 2 + 10 = 12 \]
So, \( B'(0, 12) \).
For C(3, 2):
\[ x' = 2 + 2(3 - 2) = 2 + 2 \cdot 1 = 2 + 2 = 4 \] \[ y' = 2 + 2(2 - 2) = 2 + 2 \cdot 0 = 2 + 0 = 2 \]
So, \( C'(4, 2) \).
Summary of the Dilated Triangle A'B'C'
The coordinates of the dilated triangle (assuming a scale factor of \( k = 2 \)) are:
- \( A'(8, 8) \)
- \( B'(0, 12) \)
- \( C'(4, 2) \)
If a different scale factor is desired, you can replace \( k = 2 \) with that value and recalculate accordingly.