Bot GPT 4-o mini To find the coordinates of the dilated triangle \( A'B'C' \) after dilating triangle \( ABC \) using a magnitude of 2 and a center of dilation at \( (2, 0) \), we can apply the following formula for dilation:
\[ (x', y') = (k(x - c_x) + c_x, k(y - c_y) + c_y) \]
where:
- \( (x, y) \) are the original coordinates,
- \( (c_x, c_y) \) is the center of dilation,
- \( k \) is the magnitude of dilation.
Given:
- Center of dilation \( (c_x, c_y) = (2, 0) \)
- Magnitude of dilation \( k = 2 \)
Now we will calculate the coordinates of points \( A' \), \( B' \), and \( C' \).
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For point A (1, 6): \[ A' = (2 + 2 \cdot (1 - 2), 0 + 2 \cdot (6 - 0)) = (2 + 2 \cdot (-1), 0 + 12) = (0, 12) \]
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For point B (3, 0): \[ B' = (2 + 2 \cdot (3 - 2), 0 + 2 \cdot (0 - 0)) = (2 + 2 \cdot (1), 0 + 0) = (4, 0) \]
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For point C (6, 4): \[ C' = (2 + 2 \cdot (6 - 2), 0 + 2 \cdot (4 - 0)) = (2 + 2 \cdot (4), 0 + 8) = (2 + 8, 8) = (10, 8) \]
Thus, the coordinates of the dilated triangle \( A'B'C' \) are:
- \( A' (0, 12) \)
- \( B' (4, 0) \)
- \( C' (10, 8) \)
The correct response from the options you provided is: A' (0, 12), B' (4, 0), C' (10, 8).
1 answer