To find the resulting coordinates of triangle \(ABC\) after undergoing the transformations of a 90-degree counter-clockwise rotation about the origin followed by a dilation by a factor of 12 about the origin, we proceed with the following steps:
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Rotate the points 90 degrees counter-clockwise: The transformation for a point \((x, y)\) rotated 90 degrees counter-clockwise about the origin is given by the new coordinates \((-y, x)\).
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Dilate the points by a factor of 12: After obtaining the coordinates from the rotation, we then multiply the \(x\) and \(y\) coordinates of each point by 12.
Let's denote the original coordinates of the vertices of triangle \(ABC\) as \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\).
Step-by-step transformation:
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Rotate 90 degrees counter-clockwise:
- \( A'( -y_1, x_1) \)
- \( B'( -y_2, x_2) \)
- \( C'( -y_3, x_3) \)
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Dilation by a factor of 12:
- \( A''( -12y_1, 12x_1) \)
- \( B''( -12y_2, 12x_2) \)
- \( C''( -12y_3, 12x_3) \)
Resulting Coordinates:
Assuming the original coordinates of \(A\), \(B\), and \(C\) were given or can be generated from context, we can then compute the final coordinates after the transformations.
Now, let's examine the answer choices provided:
- \(A′(−2, 0); B′(2,−3); C′(0, 1)\)
- \(A′(−4, 0); B′(4,−6); C′(0, 2)\)
- \(A′(−2, 0); B′(2, 3); C′(0,−1)\)
- \(A′(−8, 0); B′(8, 12); C′(0,−4)\)
Without explicit original coordinates, the approach is to test these transformations against assumptions about the original vertices until matching indicated coordinates are found.
Observations:
To match the transformations:
- The coordinates post-rotation should yield new meaningful coordinates when dilated by 12.
For a potential candidate: If we take point computations and calculate — one way is if we test values of vertices until an observable pattern emerges with expected characteristics of inputs — being careful to check what \(A\), \(B\), and \(C\) were prior and how they thus interact with larvae transformations.
Final Note:
We would need the original vertices of triangle ABC, and following the above transformations correctly will yield the final coordinates which could then be compared to the results provided in the options list. If vertices are available, please input, and I'll do the calculations to provide confirmation or negate the choices.
1 answer