To dilate a triangle with a center of dilation and a scale factor, you can use the following formula for each vertex:
\[ (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c)) \]
where \((x_c, y_c)\) is the center of dilation, \((x, y)\) are the coordinates of the original point, and \(k\) is the scale factor.
In this case:
- Center of dilation, \(C(2, 2)\)
- Scale factor, \(k = \frac{1}{2}\)
We will apply the dilation to each vertex of the triangle.
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Dilating point A(5, 5): \[ x' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{1}{2} \times 3 = 2 + 1.5 = 3.5 \] \[ y' = 2 + \frac{1}{2}(5 - 2) = 2 + \frac{1}{2} \times 3 = 2 + 1.5 = 3.5 \] So, the new coordinates for point A' are (3.5, 3.5).
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Dilating point B(1, 7): \[ x' = 2 + \frac{1}{2}(1 - 2) = 2 + \frac{1}{2} \times (-1) = 2 - 0.5 = 1.5 \] \[ y' = 2 + \frac{1}{2}(7 - 2) = 2 + \frac{1}{2} \times 5 = 2 + 2.5 = 4.5 \] So, the new coordinates for point B' are (1.5, 4.5).
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Dilating point C(3, 2): \[ x' = 2 + \frac{1}{2}(3 - 2) = 2 + \frac{1}{2} \times 1 = 2 + 0.5 = 2.5 \] \[ y' = 2 + \frac{1}{2}(2 - 2) = 2 + \frac{1}{2} \times 0 = 2 + 0 = 2 \] So, the new coordinates for point C' are (2.5, 2).
In conclusion, after dilating triangle ABC by a factor of \( \frac{1}{2} \) with a center at (2, 2), the coordinates of the dilated triangle A'B'C' are:
- A'(3.5, 3.5)
- B'(1.5, 4.5)
- C'(2.5, 2)