Question

Use the graph to answer the question.

graph of polygon ABCD with vertices at 0 comma 0, 5 comma 2, 5 comma negative 5, 0 comma negative 3

Determine the coordinates of polygon A′B′C′D′ if polygon ABCD is rotated 90° counterclockwise.
A′(0, 0), B′(−2, 5), C′(5, 5), D′(3, 0)
A′(0, 0), B′(−2, −5), C′(−5, 5), D′(−3, 0)
A′(0, 0), B′(−5, −2), C′(5, −5), D′(3, 0)
A′(0, 0), B′(−5, −2), C′(−5, −5), D′(0, 3)

Answer 1

To determine the coordinates of polygon \( A'B'C'D' \) after a 90° counterclockwise rotation of polygon \( ABCD \), we can apply the rotation transformation to each vertex of the original polygon.

The vertices of polygon \( ABCD \) are:
- \( A(0, 0) \)
- \( B(5, 2) \)
- \( C(5, -5) \)
- \( D(0, -3) \)

The formula for rotating a point \( (x, y) \) 90° counterclockwise about the origin is:
\[
(x', y') = (-y, x)
\]

Applying this transformation to each vertex:

1. For \( A(0, 0) \):
\[
A'(0, 0) \rightarrow A'(0, 0)
\]

2. For \( B(5, 2) \):
\[
B'(5, 2) \rightarrow B'(-2, 5)
\]

3. For \( C(5, -5) \):
\[
C'(5, -5) \rightarrow C'(5, 5)
\]

4. For \( D(0, -3) \):
\[
D'(0, -3) \rightarrow D'(3, 0)
\]

Thus, the coordinates of the rotated polygon \( A'B'C'D' \) are:
- \( A'(0, 0) \)
- \( B'(-2, 5) \)
- \( C'(5, 5) \)
- \( D'(3, 0) \)

Comparing these results with the provided options:
- The first option is \( A'(0, 0), B'(-2, 5), C'(5, 5), D'(3, 0) \), which matches our calculation.

So, the correct answer is:
**A′(0, 0), B′(−2, 5), C′(5, 5), D′(3, 0)**