To find the coordinates of triangle \( A''B''C'' \) after the given transformations, we will first apply the transformation rules step by step and record the results.
Step 1: Apply the first transformation
The first transformation is defined by the rule \( (x, y) \to (x + 4, y - 6) \).
Starting with the original coordinates of triangle ABC:
- A(-8, 4)
- B(-5, 8)
- C(-3, 3)
Applying the transformation to each vertex:
For A: \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]
For B: \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]
For C: \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]
So after the first transformation, the coordinates of triangle \( A'B'C' \) are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)
Step 2: Apply the second transformation
The second transformation is defined by the rule \( (x, y) \to (x + 5, y - 3) \).
Now we will apply this transformation to each of the new vertices:
For A': \[ A'' = (-4 + 5, -2 - 3) = (1, -5) \]
For B': \[ B'' = (-1 + 5, 2 - 3) = (4, -1) \]
For C': \[ C'' = (1 + 5, -3 - 3) = (6, -6) \]
So after the second transformation, the coordinates of triangle \( A''B''C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)
Step 3: Determine the quadrant
Next, we need to predict which quadrant triangle \( A''B''C'' \) will be in. The key point is that all coordinates of the vertices \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \) have a positive x-coordinate and a negative y-coordinate.
Conclusion
Since all new vertices have \( x > 0 \) and \( y < 0 \), this places triangle \( A''B''C'' \) in the fourth quadrant of the Cartesian plane.
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