To find the coordinates of \( \Delta A"B"C" \) after the transformations, we will apply each transformation step by step.
Initial Coordinates
The original coordinates of the triangle \( \Delta ABC \) are:
- \( A(-8, 4) \)
- \( B(-5, 8) \)
- \( C(-3, 3) \)
First Transformation
The first transformation is given by the rule \( (x, y) \to (x + 4, y - 6) \).
We apply this to each vertex of \( \Delta ABC \):
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For \( A' \): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]
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For \( B' \): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]
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For \( C' \): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]
So, the coordinates of \( \Delta A'B'C' \) are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)
Second Transformation
The second transformation is given by the rule \( (x, y) \to (x + 5, y - 3) \).
We will now apply this transformation to each vertex of \( \Delta A'B'C' \):
-
For \( A'' \): \[ A'' = (-4 + 5, -2 - 3) = (1, -5) \]
-
For \( B'' \): \[ B'' = (-1 + 5, 2 - 3) = (4, -1) \]
-
For \( C'' \): \[ C'' = (1 + 5, -3 - 3) = (6, -6) \]
Thus, the coordinates of \( \Delta A"B"C" \) after both transformations are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
Composition Rule
To create the composition transformation from the original coordinates \( (x, y) \) to \( (x + 9, y - 9) \) (adding both transformations together):
- The total transformation can be viewed as: \[ (x, y) \to (x + 4 + 5, y - 6 - 3) \to (x + 9, y - 9) \]
Quadrant Prediction
Now, let's analyze the final coordinates \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \):
- All these points have a positive x-coordinate and a negative y-coordinate. This means:
- \( x > 0 \) and \( y < 0 \)
Therefore, the points lie in Quadrant IV.
Conclusion
The coordinates of \( \Delta A"B"C" \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
The triangle \( \Delta A"B"C" \) will be in Quadrant IV after the transformations.