The coordinates of △ABC△*ABC*​ are A(−3,−3)*A*(−3,−3)​​, B(−2,−1)*B*(−2,−1)​​, and C(−1,−6)*C*(−1,−6)​. △ABC△*ABC*​ is rotated 90°90°​ counterclockwise about the origin and then translated by the translation vector ⟨−6,−3⟩⟨−6,−3⟩​​. 

a) What are the coordinates of A′′*A*′′​?

To find the coordinates of \( A'' \), the final position of point \( A \) after the transformations, we need to follow these steps:

1. **Rotate \( A \) by \( 90^\circ \) counterclockwise about the origin.**
   
   Given point \( A(-3, -3) \), when we rotate any point \( (x, y) \) by \( 90^\circ \) counterclockwise about the origin, the new coordinates \( (x', y') \) become:
   \[
   (x', y') = (-y, x)
   \]
   
   Applying this to point \( A(-3, -3) \):
   \[
   A' = (-(-3), -3) = (3, -3)
   \]

2. **Translate \( A' \) by the vector \( \langle -6, -3 \rangle \).**

   To translate a point by a vector, we add the vector components to the coordinates of the point. For \( A'(3, -3) \) and the translation vector \( \langle -6, -3 \rangle \):
   \[
   A'' = (3 + (-6), -3 + (-3)) = (-3, -6)
   \]

### Final Answer:
The coordinates of \( A'' \) are \( (-3, -6) \).

Would you like more details on any of the steps?

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Here are five related questions that build on this solution:

1. What are the coordinates of \( B'' \) after applying the same transformations?
2. What are the coordinates of \( C'' \) after the same transformations?
3. How would the process differ if the rotation were \( 180^\circ \) instead of \( 90^\circ \)?
4. How can we determine the overall effect of a rotation followed by a translation on any point in the plane?
5. What transformation matrix could represent the combined rotation and translation?

**Tip:** When rotating points around the origin, use the general rotation formulas for clarity and avoid errors with negative coordinates.

The coordinates of △ABC are A(-3,-3), B(-2,-1), and C(-1,-6). △ABC is rotated 90° counterclockwise about the origin and then translated by the vector ⟨-6,-3⟩. What are the coordinates of A''?

Explore how to find the new coordinates of a point on triangle ABC after a 90° counterclockwise rotation about the origin followed by a translation using vector ⟨-6,-3⟩. Learn step-by-step solutions for transformations in coordinate geometry, suitable for Grades 8-10.

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