https://file.aiquickdraw.com/mathbot/file/1726404252695uz3agouv.jpg

The problem involves two transformations on point \( A(8, -8) \): a translation followed by a rotation.

### Translation
Given the translation vector \( T = \left(-2, 9\right) \):
\[ A'(x', y') = (x + \Delta x, y + \Delta y) \]
\[ A'(x', y') = (8 - 2, -8 + 9) \]
\[ A'(x', y') = (6, 1) \]

### Rotation
After translating, the new point \( A' \) is \( (6, 1) \). We need to rotate this point 90° counterclockwise around the origin (0, 0). The rotation matrix for 90° counterclockwise is:
\[ R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

The rotation is applied as follows:
\[ A''(x'', y'') = R \cdot A' \]
\[ A''(x'', y'') = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 6 \\ 1 \end{bmatrix} \]
\[ A''(x'', y'') = \begin{bmatrix} 0 \cdot 6 + (-1) \cdot 1 \\ 1 \cdot 6 + 0 \cdot 1 \end{bmatrix} \]
\[ A''(x'', y'') = \begin{bmatrix} -1 \\ 6 \end{bmatrix} \]

### Conclusion
The final coordinates of the point after translation and rotation are \( (-1, 6) \).

Would you like more details on any part of this process, or do you have any further questions?

**Further Queries:**
1. Would you like to understand more about the geometric interpretation of these transformations?
2. Are you interested in learning about rotation matrices for other angles?
3. Do you want to explore translations in other contexts or dimensions?
4. Need to apply similar transformations to a set of points or a shape?
5. How can these transformations be useful in practical applications?

**Tip:** When dealing with transformations, it's helpful to sketch the points and vectors on graph paper to visually track the movement and rotation.

Titik A(8, -8) jika ditranslasi oleh T = (-2, 9) dilanjutkan rotasi dengan pusat (0,0) searah jarum jam sejauh 90° akan menghasilkan bayangan dengan koordinat....

Learn how to apply translation and 90° clockwise rotation to point A(8, -8). This problem covers key geometric concepts, including translation and rotation transformations in the Cartesian plane. Suitable for students in Grades 9-11.

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