Question

Plot and state the points of new house from chapter 2 after countercloclwise rotation 180 degrees about the (-2,1)(0,1)(0,3)(-1,4)(-2,3)

Answer 1

To rotate a point \( (x, y) \) counterclockwise by 180 degrees about a center point \( (h, k) \), you can use the following formula:

\[
(x', y') = (h - (x - h), k - (y - k)) = (2h - x, 2k - y)
\]

For the points you've provided, the center of rotation is \( (-2, 1) \). Now let's apply the rotation formula to each of the points given:

1. **Point (0, 1)**:
\[
(x', y') = (2(-2) - 0, 2(1) - 1) = (-4, 1)
\]

2. **Point (0, 3)**:
\[
(x', y') = (2(-2) - 0, 2(1) - 3) = (-4, -1)
\]

3. **Point (-1, 4)**:
\[
(x', y') = (2(-2) - (-1), 2(1) - 4) = (-3, -2)
\]

4. **Point (-2, 3)**:
\[
(x', y') = (2(-2) - (-2), 2(1) - 3) = (-2, -1)
\]

5. **Point (-2, 1)**:
\[
(x', y') = (2(-2) - (-2), 2(1) - 1) = (-2, 1)
\]

Now let’s summarize the points after the 180-degree rotation about (-2, 1):

1. **(0, 1)** becomes **(-4, 1)**
2. **(0, 3)** becomes **(-4, -1)**
3. **(-1, 4)** becomes **(-3, -2)**
4. **(-2, 3)** becomes **(-2, -1)**
5. **(-2, 1)** remains **(-2, 1)**

Here are the new points after the rotation:
- Point \( (-4, 1) \)
- Point \( (-4, -1) \)
- Point \( (-3, -2) \)
- Point \( (-2, -1) \)
- Point \( (-2, 1) \)

You can plot these points on a coordinate grid to visualize the transformation.