Let's first find the coordinates of \(\Delta A'B'C'\) after applying the first transformation to \(\Delta ABC\).
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Original Coordinates:
- \(A(-8, 4)\)
- \(B(-5, 8)\)
- \(C(-3, 3)\)
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Transformation 1: \((x, y) \to (x + 4, y - 6)\)
Applying this transformation to each point:
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For point \(A\): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]
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For point \(B\): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]
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For point \(C\): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]
Now the coordinates of \(\Delta A' B' C'\) are:
- \(A'(-4, -2)\)
- \(B'(-1, 2)\)
- \(C'(1, -3)\)
- Transformation 2: \((x, y) \to (x + 5, y - 3)\)
Next, we apply this transformation to each point of \(\Delta A'B'C'\):
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For point \(A'\): \[ A'' = (-4 + 5, -2 - 3) = (1, -5) \]
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For point \(B'\): \[ B'' = (-1 + 5, 2 - 3) = (4, -1) \]
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For point \(C'\): \[ C'' = (1 + 5, -3 - 3) = (6, -6) \]
The coordinates of \(\Delta A''B''C''\) are:
- \(A''(1, -5)\)
- \(B''(4, -1)\)
- \(C''(6, -6)\)
- Create the Rule for the Composition Transformation:
The overall transformation is a combination of both transformations:
- First Transformation: \((x, y) \to (x + 4, y - 6)\)
- Second Transformation: \((x, y) \to (x + 5, y - 3)\)
To carry out the composition:
- The combined transformation can be written as:
\[ (x, y) \to (x + 4 + 5, y - 6 - 3) = (x + 9, y - 9) \]
- Predicting the Quadrant:
Now let's analyze the final coordinates:
- \(A''(1, -5)\), \(B''(4, -1)\), and \(C''(6, -6)\).
All points have a positive x-coordinate and a negative y-coordinate. Thus, they are located in Quadrant IV, where \(x > 0\) and \(y < 0\).
Final Result:
- The coordinates of \(\Delta A''B''C''\) are \(A''(1, -5)\), \(B''(4, -1)\), and \(C''(6, -6)\).
- The transformation rule is \((x, y) \to (x + 9, y - 9)\).
- The resulting image will be in Quadrant IV.
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