Question

Polygon ABCD is rotated to get polygon A′B′C′D′.

Graph of polygon ABCD in quadrant 1 with point A at 1 comma 5, point B at 6 comma 8, point C at 6 comma 4, and point D at 4 comma 3 and polygon A prime B prime C prime D prime in quadrant 4 with point A prime at 5 comma negative 1, point B prime at 8 comma negative 6, point C prime at 4 comma negative 6, and point D prime at 3 comma negative 4

Determine the direction and angle of rotation.

270° clockwise rotation
180° counterclockwise rotation
90° counterclockwise rotation
90° clockwise

Answer 1

To determine the direction and angle of rotation from polygon ABCD to polygon A′B′C′D′, we need to compare the coordinates of the original polygon with those of the rotated polygon.

The coordinates for polygon ABCD are:
- A(1, 5)
- B(6, 8)
- C(6, 4)
- D(4, 3)

The coordinates for polygon A′B′C′D′ are:
- A′(5, -1)
- B′(8, -6)
- C′(4, -6)
- D′(3, -4)

When a point \((x, y)\) is rotated 90° clockwise around the origin, the new coordinates become \((y, -x)\). For a 270° clockwise rotation (which is equivalent to 90° counterclockwise), the coordinates become \((-y, x)\).

Let's check the points one by one to see which rotation fits:

1. **Point A(1, 5)** to A′(5, -1):
- For 90° clockwise: \( (5, -1) \) is not \((5, -1)\).
- For 270° clockwise:
\[
(-5, 1) \Rightarrow (5, -1) \text{ (not matches)}
\]
- For 180°:
\[
(-1, -5) \Rightarrow (5, -1) \text{ (not matches)}
\]
- For 90° counterclockwise:
\[
(-5, 1) \Rightarrow (5, -1) \text{ (not a match)}
\]

Now let's try a 90° clockwise rotation for all points:

1. **A(1, 5)**:
- \( (5, -1) \) → \( 90° \text{ clockwise } \) so correct.

2. **B(6, 8)**:
- \( (8, -6) \) → \( 90° \text{ clockwise } \) correct.

3. **C(6, 4)**:
- \( (4, -6) \) → \( 90° \text{ clockwise } \) correct.

4. **D(4, 3)**:
- \( (3, -4) \) → \( 90° \text{ clockwise } \) correct.

Since all the points successfully map through the 90° clockwise rotation.

### Conclusion:
The correct answer is **90° clockwise rotation**.