To find the coordinates of ΔA"B"C" after the transformations and determine the quadrant in which they will be located, let's follow the steps outlined in the problem.
Step 1: Apply the First Transformation
The first transformation is given by the rule \( (x,y) \to (x+4, y-6) \).
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Transform A(−8,4):
- \( A' = (-8 + 4, 4 - 6) = (-4, -2) \)
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Transform B(−5,8):
- \( B' = (-5 + 4, 8 - 6) = (-1, 2) \)
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Transform C(−3,3):
- \( C' = (-3 + 4, 3 - 6) = (1, -3) \)
So, after the first transformation, the coordinates of ΔA'B'C' are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)
Step 2: Apply the Second Transformation
The second transformation is given by the rule \( (x,y) \to (x+5, y-3) \).
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Transform A'(-4, -2):
- \( A'' = (-4 + 5, -2 - 3) = (1, -5) \)
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Transform B'(-1, 2):
- \( B'' = (-1 + 5, 2 - 3) = (4, -1) \)
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Transform C'(1, -3):
- \( C'' = (1 + 5, -3 - 3) = (6, -6) \)
After the second transformation, the coordinates of ΔA"B"C" are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
Step 3: Determine the Quadrant
Now, let's analyze the coordinates of points A'', B'', and C'' to determine the quadrant in which ΔA"B"C" lies.
- Point A''(1, -5): x is positive, y is negative → Quadrant IV
- Point B''(4, -1): x is positive, y is negative → Quadrant IV
- Point C''(6, -6): x is positive, y is negative → Quadrant IV
All three points are in the fourth quadrant (Quadrant IV).
Conclusion
The coordinates of ΔA"B"C" after the transformations are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
The image will be in Quadrant IV after completing the transformations.