A(6,0) first <-3,3> then reflect over y axis last rotate the given point 90 degrees about P(1,-2)

Let's solve this step by step.

### 1. **Apply the translation**:  
The first transformation is a translation by the vector \(\langle -3, 3 \rangle\).  
The point \(A(6, 0)\) will be moved as follows:
\[
(6, 0) + (-3, 3) = (6 - 3, 0 + 3) = (3, 3)
\]
After the translation, the new point is \(A'(3, 3)\).

### 2. **Reflect over the y-axis**:  
Next, reflect the point \(A'(3, 3)\) over the y-axis.  
When reflecting a point \((x, y)\) over the y-axis, the \(x\)-coordinate changes sign while the \(y\)-coordinate remains the same:
\[
(3, 3) \to (-3, 3)
\]
After the reflection, the point is \(A''(-3, 3)\).

### 3. **Rotate 90 degrees counterclockwise about \(P(1, -2)\)**:
Finally, we need to rotate the point \(A''(-3, 3)\) 90 degrees counterclockwise around \(P(1, -2)\).

To rotate a point \((x, y)\) around another point \((x_p, y_p)\) by 90 degrees counterclockwise:
1. Translate the point so that \(P(1, -2)\) becomes the origin:
   \[
   x' = x - x_p \quad \text{and} \quad y' = y - y_p
   \]
   For \(A''(-3, 3)\):
   \[
   x' = -3 - 1 = -4 \quad \text{and} \quad y' = 3 - (-2) = 5
   \]
   So, the translated point is \((-4, 5)\).

2. Perform the 90-degree rotation counterclockwise:
   \[
   (x', y') \to (-y', x')
   \]
   So, \((-4, 5) \to (-5, -4)\).

3. Translate the point back to the original position of \(P(1, -2)\):
   \[
   x_{\text{new}} = x' + x_p \quad \text{and} \quad y_{\text{new}} = y' + y_p
   \]
   \[
   x_{\text{new}} = -5 + 1 = -4 \quad \text{and} \quad y_{\text{new}} = -4 + (-2) = -6
   \]
   Therefore, the final point is \((-4, -6)\).

### Final Answer:  
After performing all the transformations, the final position of the point is \(\boxed{(-4, -6)}\).

Would you like any further clarifications on these steps?

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Here are 5 related questions to extend your understanding:

1. What happens if the reflection is done over the x-axis instead of the y-axis?
2. How would the rotation change if it was done clockwise instead of counterclockwise?
3. What is the general formula for rotating any point around an arbitrary point by 180 degrees?
4. How would the translation change if the vector was \(\langle 3, -3 \rangle\) instead?
5. How do you rotate a point about the origin by an angle that is not 90 degrees?

**Tip**: When reflecting points over the x-axis or y-axis, only one coordinate changes sign!

Solve the geometric problem where the point A(6,0) is first translated by <-3,3>, reflected over the y-axis, and finally rotated 90 degrees counterclockwise about the point P(1,-2). Learn step-by-step methods using translation, reflection, and rotation formulas. Suitable for students in Grades 9-11.

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