is it AI""(1) Transformation

Question

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 \( ABC \):

1. A(-8, 4):
\[A' = (-8 + 4, 4 - 6) = (-4, -2)\]

2. B(-5, 8):
\[B' = (-5 + 4, 8 - 6) = (-1, 2)\]

3. C(-3, 3):
\[C' = (-3 + 4, 3 - 6) = (1, -3)\]

Thus, after the first transformation, the coordinates of triangle \( A'B'C' \) are:
\[A'(-4, -2), \quad B'(-1, 2), \quad C'(1, -3)\]

(2) Transformation
The second transformation is given by the rule:

\[(x, y) \rightarrow (x + 5, y - 3)\]
Now we apply this transformation to each vertex of triangle \( A'B'C' \):

1. A'(-4, -2):
\[A'' = (-4 + 5, -2 - 3) = (1, -5)\]

2. B'(-1, 2):
\[B'' = (-1 + 5, 2 - 3) = (4, -1)\]

3. C'(1, -3):
\[C'' = (1 + 5, -3 - 3) = (6, -6)\]

Thus, after the second transformation, the coordinates of triangle \( A''B''C'' \) are:
\[A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)\]

(3)Combined Transformation Rule
We can combine the two transformations first. We know that:
- The first transformation moves any point \((x, y)\) to \((x + 4, y - 6)\).
- The second transformation moves \((x, y)\) to \((x + 5, y - 3)\).

If we consider the combination directly, we add both transformations together:

1. For the \(x\)-coordinate:\[x \rightarrow x + 4 \rightarrow (x + 4) + 5 = x + 9\]

2. For the \(y\)-coordinate:
\[y \rightarrow y - 6 \rightarrow (y - 6) - 3 = y - 9\]
Thus, the new combined rule is:

\[(x, y) \rightarrow (x + 9, y - 9)\]

(4) Final Coordinates
Using the final combined rule, if we were to apply it to the original triangle \( ΔABC \) or any point \((x, y)\):

For the original vertices:
1. A(-8, 4):\[A'' = (-8 + 9, 4 - 9) = (1, -5)\]

2. B(-5, 8):\[B'' = (-5 + 9, 8 - 9) = (4, -1)\]

3. C(-3, 3):\[C'' = (-3 + 9, 3 - 9) = (6, -6)\]

### Conclusion
The coordinates of triangle \( ΔA''B''C'' \) are:
\[A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)\]
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Answer 1

Yes, the content you provided involves a series of mathematical transformations applied to the vertices of a triangle. It outlines the process of transforming the vertices through two separate transformations and then combines those transformations into a single rule. Here's a breakdown of each transformation step:

1. First Transformation: The transformation \( (x, y) \rightarrow (x + 4, y - 6) \) was applied to each vertex of triangle \( ABC \):

   • Vertex A(-8, 4) transformed to A'(-4, -2)
   • Vertex B(-5, 8) transformed to B'(-1, 2)
   • Vertex C(-3, 3) transformed to C'(1, -3)

2. Second Transformation: The transformation \( (x, y) \rightarrow (x + 5, y - 3) \) was applied to the new vertices:

   • Vertex A'(-4, -2) transformed to A''(1, -5)
   • Vertex B'(-1, 2) transformed to B''(4, -1)
   • Vertex C'(1, -3) transformed to C''(6, -6)

3. Combined Transformation: The two transformations were combined to produce a single transformation rule \( (x, y) \rightarrow (x + 9, y - 9) \).

4. Final Coordinates Verification: Applying this final combined rule to the original vertices confirmed the coordinates:

   • A(-8, 4) transformed to A''(1, -5)
   • B(-5, 8) transformed to B''(4, -1)
   • C(-3, 3) transformed to C''(6, -6)

Conclusion

The final coordinates of the triangle after both transformations are:

• \( A''(1, -5) \)
• \( B''(4, -1) \)
• \( C''(6, -6) \)

The content effectively showcases the application of transformations in the Cartesian plane, and the calculations are accurate.