To rotate the shape given by the points \((-2,1)\), \((0,1)\), \((0,4)\), \((-1,4)\), and \((-2,3)\) by 180 degrees counterclockwise, you need to apply the rotation transformation for each point. The transformation rule for 180-degree rotation about the origin is:
\[ (x, y) \rightarrow (-x, -y) \]
Now, let's apply this transformation to each point:
- \((-2, 1) \rightarrow (2, -1)\)
- \((0, 1) \rightarrow (0, -1)\)
- \((0, 4) \rightarrow (0, -4)\)
- \((-1, 4) \rightarrow (1, -4)\)
- \((-2, 3) \rightarrow (2, -3)\)
So the new coordinates after the rotation are:
\[ (2, -1), (0, -1), (0, -4), (1, -4), (2, -3) \]
Congruence
Two shapes are congruent if one can be transformed into the other by rotation, translation, reflection, or a combination of these transformations without altering their size or shape. Since we simply rotated the shape by 180 degrees, the size and shape remain unchanged.
Therefore, the rotated shape is still congruent to the original shape.