To determine the coordinates of the dilated triangle \(X'Y'Z'\) from triangle \(XYZ\) using a scale factor of 2 and a center of dilation at the point \((2,0)\), we can follow these steps:
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Identify the Formula for Dilation: The formula for dilation of a point \((x, y)\) with respect to a center of dilation \((c, d)\) and a scale factor \(k\) is: \[ (x', y') = (c + k(x - c), d + k(y - d)) \] Here, \(c = 2\) and \(d = 0\), while \(k = 2\).
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Apply Dilation to Each Point: We need the original coordinates of points \(X\), \(Y\), and \(Z\) to compute their dilated counterparts. Unfortunately, these original coordinates are not provided in the question, but we can assume some points that can be dilated to help explain the process.
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Example: Let's say we had:
- \(X(1, 1)\)
- \(Y(2, 0)\)
- \(Z(3, 1)\)
We can apply the dilation to each of these points:
- For point \(X (1, 1)\): \[ X' = \left(2 + 2(1 - 2), 0 + 2(1 - 0)\right) = (2 - 2, 0 + 2) = (0, 2) \]
- For point \(Y (2, 0)\): \[ Y' = \left(2 + 2(2 - 2), 0 + 2(0 - 0)\right) = (2 + 0, 0 + 0) = (2, 0) \]
- For point \(Z (3, 1)\): \[ Z' = \left(2 + 2(3 - 2), 0 + 2(1 - 0)\right) = (2 + 2, 0 + 2) = (4, 2) \]
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Conclusion: After applying dilation to the assumed points, the new coordinates would be \(X'(0, 2)\), \(Y'(2, 0)\), \(Z'(4, 2)\).
Since the coordinates for triangle \(XYZ\) are not provided, we have to use the options given in the question to form the triangle's new points. However, from the options, it looks like you have:
- Option 1: \(X'(2,2), Y'(4,0), Z'(4.5,1)\)
- Option 2: \(X'(6,4), Y'(10,0), Z'(11,4)\)
- Option 3: \(X'(2,8), Y'(10,0), Z'(12,4)\)
- Option 4: \(X'(2,6), Y'(8,0), Z'(9.5,3)\)
Without the original points, I cannot directly compute the expected dilated triangle coordinates. You will need the original triangle coordinates \(X\), \(Y\), and \(Z\) to verify which option is correct.
If you provide the coordinates of triangle \(XYZ\), I can help you find the exact coordinates of the dilated triangle \(X'Y'Z'\).