To find the coordinates of \(\Delta A"B"C"\) after the transformations, we will follow these steps:
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Apply the first transformation: \((x,y) \to (x+4, y-6)\) to the vertices of \(\Delta ABC\).
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For \(A(-8, 4)\): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]
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For \(B(-5, 8)\): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]
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For \(C(-3, 3)\): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]
After the first transformation, the coordinates of \(\Delta A'B'C'\) are:
- \(A'(-4, -2)\)
- \(B'(-1, 2)\)
- \(C'(1, -3)\)
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Apply the second transformation: \((x,y) \to (x+5, y-3)\) to the coordinates of \(\Delta A'B'C'\).
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For \(A'(-4, -2)\): \[ A" = (-4 + 5, -2 - 3) = (1, -5) \]
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For \(B'(-1, 2)\): \[ B" = (-1 + 5, 2 - 3) = (4, -1) \]
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For \(C'(1, -3)\): \[ C" = (1 + 5, -3 - 3) = (6, -6) \]
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Coordinates of \(\Delta A"B"C"\):
- \(A"(1, -5)\)
- \(B"(4, -1)\)
- \(C"(6, -6)\)
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Determine the quadrant:
- All final coordinates \((1, -5)\), \((4, -1)\), and \((6, -6)\) have positive \(x\)-values and negative \(y\)-values. Therefore, they are all in the fourth quadrant.
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Create the rule for the composition transformation:
- The combined transformation can be represented as: \[ (x, y) \to (x + 4, y - 6) \quad \text{then} \quad (x + 5, y - 3) \]
- If we combine these transformations, we get: \[ (x, y) \to ((x + 4) + 5, (y - 6) - 3) = (x + 9, y - 9) \]
So, the composite transformation rule is: \[ (x, y) \to (x + 9, y - 9) \]
In conclusion, the final coordinates of \(\Delta A"B"C"\) are \(A"(1, -5)\), \(B"(4, -1)\), and \(C"(6, -6)\), and the transformed triangle is located in the fourth quadrant.