Question

Your house is located at: A (-2,-1); B( 0,-1); C(0,-3); D(1,-4); E(2,-3)
what are the new points if you rotate the house 180 degrees counterclockwise about the origin? are the houses congruent?

Answer 1

To rotate a point \((x, y)\) 180 degrees counterclockwise about the origin, the new coordinates will be \((-x, -y)\).

Now, let's apply this transformation to each of the points A, B, C, D, and E:

1. **Point A (-2, -1)**:
\[
A' = (-(-2), -(-1)) = (2, 1)
\]

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

3. **Point C (0, -3)**:
\[
C' = (-0, -(-3)) = (0, 3)
\]

4. **Point D (1, -4)**:
\[
D' = (-1, -(-4)) = (-1, 4)
\]

5. **Point E (2, -3)**:
\[
E' = (-2, -(-3)) = (-2, 3)
\]

Thus, the new points after rotating the house 180 degrees counterclockwise about the origin are:
- \(A' = (2, 1)\)
- \(B' = (0, 1)\)
- \(C' = (0, 3)\)
- \(D' = (-1, 4)\)
- \(E' = (-2, 3)\)

### Congruence
Two shapes are congruent if they have the same size and shape, which means their corresponding sides and angles are equal. Since a 180-degree rotation does not change the size or shape of the figure, the new points represent a figure that is congruent to the original one.

**Conclusion**: The new points after rotation are \( A' (2, 1), B' (0, 1), C' (0, 3), D' (-1, 4), E' (-2, 3) \), and the original and rotated shapes are congruent.