To find the coordinates of the transformed triangle \( A''B''C'' \) and determine in which quadrant it will lie, we need to apply the transformations step-by-step.
Step 1: First Transformation
The first transformation is given by the rule:
\[ (x, y) \rightarrow (x + 4, y - 6) \]
Let's apply this transformation to each vertex of triangle \( ABC \):
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Point A: \[ A(-8, 4) \rightarrow (-8 + 4, 4 - 6) = (-4, -2) \]
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Point B: \[ B(-5, 8) \rightarrow (-5 + 4, 8 - 6) = (-1, 2) \]
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Point C: \[ C(-3, 3) \rightarrow (-3 + 4, 3 - 6) = (1, -3) \]
So, after the first transformation, the coordinates of triangle \( A'B'C' \) are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)
Step 2: Second Transformation
The second transformation is defined by the rule:
\[ (x, y) \rightarrow (x + 5, y - 3) \]
Now, we apply this transformation to each point of triangle \( A'B'C' \):
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Point A': \[ A'(-4, -2) \rightarrow (-4 + 5, -2 - 3) = (1, -5) \]
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Point B': \[ B'(-1, 2) \rightarrow (-1 + 5, 2 - 3) = (4, -1) \]
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Point C': \[ C'(1, -3) \rightarrow (1 + 5, -3 - 3) = (6, -6) \]
So, after the second transformation, the coordinates of triangle \( A''B''C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
Step 3: Predicting the Quadrant
Now, we need to determine the quadrant where triangle \( A''B''C'' \) is located.
- Point A''(1, -5): \( x > 0 \) and \( y < 0 \) (Quadrant IV)
- Point B''(4, -1): \( x > 0 \) and \( y < 0 \) (Quadrant IV)
- Point C''(6, -6): \( x > 0 \) and \( y < 0 \) (Quadrant IV)
All three points are in Quadrant IV.
Conclusion
The coordinates of triangle \( A''B''C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
After the transformations, triangle \( A''B''C'' \) will be located entirely in Quadrant IV.
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