The point (-4,1) is rotated 180 degrees counterclockwise using center (-3,0). What are the coordinates of the image?

Select the correct choice.

A


(-5,-2)

B


(-4,-1)

C


(-2,-1)

D


(4,-1)

To find the coordinates of the image of point \((-4, 1)\) after a 180-degree counterclockwise rotation around the center \((-3, 0)\), we can follow these steps:

### Step-by-Step Process:
1. **Translate the points to the origin**:  
   To make the rotation easier, we translate the center of rotation \((-3, 0)\) to the origin. This means shifting both the point \((-4, 1)\) and the center by adding \((3, 0)\) to both points:
   \[
   (-4, 1) + (3, 0) = (-1, 1)
   \]
   Now, the point is \((-1, 1)\) relative to the origin.

2. **Perform the 180-degree rotation**:  
   For a 180-degree counterclockwise rotation, the coordinates \((x, y)\) become \((-x, -y)\). Applying this to the translated point:
   \[
   (-1, 1) \rightarrow (1, -1)
   \]

3. **Translate back to the original center**:  
   Now, we translate the point back by subtracting \((3, 0)\):
   \[
   (1, -1) + (-3, 0) = (-2, -1)
   \]

Thus, the coordinates of the image after the rotation are \((-2, -1)\).

### Correct Choice:  
The correct answer is:
**C) (-2, -1)**

Would you like more details on the process? Here are some additional related questions:

1. What happens when you rotate a point by 90 degrees counterclockwise?
2. How would the coordinates change for a 180-degree rotation about the origin?
3. What is the formula for translating a point before and after a rotation?
4. How do you determine the center of rotation for a geometric figure?
5. Can you apply this rotation to a 3D point?

**Tip**: When performing rotations, always remember to translate the point relative to the center of rotation for easier calculations!

The point (-4,1) is rotated 180 degrees counterclockwise using center (-3,0). What are the coordinates of the image?

Understand how to find the image of point (-4, 1) after a 180-degree counterclockwise rotation around center (-3, 0). Learn the step-by-step process, including translation, rotation, and translation back, using the rotation transformation formula. Suitable for students in Grades 8-10.

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