Jenny’s art project requires her to rotate point *A*(6, 4) 180° counterclockwise about the point *C*(–1, 3).What are the coordinates of *A*’?

A. *A*’(–2, –8)

B. *A*’(7, 1)

C. *A*’(8, –2)

D. *A*’(–8, 2)


To find the coordinates of point \( A' \) after rotating point \( A(6, 4) \) by 180° counterclockwise about the point \( C(-1, 3) \), follow these steps:

### Step 1: Translate the points so that \( C \) is at the origin
First, we translate point \( A(6, 4) \) by moving it relative to \( C(-1, 3) \), effectively treating \( C \) as the origin.

1. Subtract the coordinates of \( C \) from \( A \):
   \[
   (6 - (-1), 4 - 3) = (6 + 1, 1) = (7, 1)
   \]

So, after translation, the coordinates of \( A \) relative to \( C \) are \( (7, 1) \).

### Step 2: Apply a 180° rotation
A 180° rotation about the origin changes \( (x, y) \) to \( (-x, -y) \).

1. Rotate \( (7, 1) \) by 180°:
   \[
   (-7, -1)
   \]

### Step 3: Translate back to the original position
Now, translate \( (-7, -1) \) back by adding the coordinates of \( C \) to return to the original coordinate system:

\[
(-7 + (-1), -1 + 3) = (-8, 2)
\]

### Conclusion
The coordinates of \( A' \) after a 180° rotation about \( C(-1, 3) \) are:

**D. \( A'(-8, 2) \)**

Jenny’s art project requires her to rotate point A(6, 4) 180° counterclockwise about the point C(–1, 3). What are the coordinates of A'?

Explore how to find the coordinates of A' after a 180° counterclockwise rotation of point A(6, 4) around point C(–1, 3). Learn to use translation and rotation transformations with a step-by-step solution. Suitable for Grades 9-10.

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