Question

The vertices of figure BCDE have coordinates B(1, 3), C(5, 3), D(5, 1), and E(1, 1) . The vertices of figure B′C′D′E′ have coordinates B′(3, −1), C′(3, −5), D′(1, −5), and E′(1, −1) .

Which transformation of​ figure BCDE ​produces figure B′C′D′E′ ​ ?


Responses

a 90° clockwise rotation about the origin
a 90° clockwise rotation about the origin

a translation 8 units upward
a translation 8 units upward

a translation 8 units downward
a translation 8 units downward

a 90° counterclockwise rotation about the origin
a 90° counterclockwise rotation about the origin

Answer 1

To determine which transformation of figure BCDE produces figure B′C′D′E′, we can analyze the coordinates of both figures.

**Coordinates of BCDE:**
- B(1, 3)
- C(5, 3)
- D(5, 1)
- E(1, 1)

**Coordinates of B′C′D′E′:**
- B′(3, −1)
- C′(3, −5)
- D′(1, −5)
- E′(1, −1)

Now, let's analyze the transformation step by step:

1. **Translation Check:**
- If we translate BCDE downwards by 8 units, we add (-8) to the y-coordinates of all points in BCDE:
- B(1, 3) → (1, 3 - 8) = (1, -5)
- C(5, 3) → (5, 3 - 8) = (5, -5)
- D(5, 1) → (5, 1 - 8) = (5, -7)
- E(1, 1) → (1, 1 - 8) = (1, -7)
- This does not match.

- If we translate BCDE upwards by 8 units, we add 8 to the y-coordinates of all points in BCDE:
- B(1, 3) → (1, 3 + 8) = (1, 11)
- C(5, 3) → (5, 3 + 8) = (5, 11)
- D(5, 1) → (5, 1 + 8) = (5, 9)
- E(1, 1) → (1, 1 + 8) = (1, 9)
- This does not match either.

2. **Check for Rotations:**
- To perform a 90° clockwise rotation about the origin, we transform the point (x, y) to (y, -x):
- B(1, 3) → (3, -1)
- C(5, 3) → (3, -5)
- D(5, 1) → (1, -5)
- E(1, 1) → (1, -1)
- This matches exactly with B′C′D′E′.

Based on the analysis above, the transformation that produces figure B′C′D′E′ from figure BCDE is:

**A 90° clockwise rotation about the origin.**