Question

Use △BCD in the figure shown.

The graph with X-coordinate marks -3, 0, 3 and Y-coordinate mark -3, 0, 3. The triangle BCD with coordinates B at (-3, 3), C at (4, 1), D at (0, -2).

Choose the correct drop-down menu item in each.

The image of (Rx-axis ∘ Ry-axis)(△BCD)
Choose...
the image r(180°, O) (△BCD).

The image of (Ry-axis ∘ Rx-axis)(△BCD)
Choose...
the image r(180°, O) (△BCD).

Answer 1

To determine the results of the transformations applied to triangle BCD, we first need to understand what each transformation does.

### Transformations:
1. **Rx-axis**: This represents a reflection across the x-axis, which changes the y-coordinates of the points by taking their negatives.
2. **Ry-axis**: This represents a reflection across the y-axis, which changes the x-coordinates of the points by taking their negatives.
3. **r(180°, O)**: This represents a rotation of 180 degrees around the origin, which negates both the x and y-coordinates of the points.

### Coordinates of Triangle BCD:
- B = (-3, 3)
- C = (4, 1)
- D = (0, -2)

### Applying the transformations:

1. **(Rx-axis ∘ Ry-axis)(△BCD)**:
- First, reflect across the y-axis (Ry-axis):
- B = (3, 3)
- C = (-4, 1)
- D = (0, -2)
- Now reflect this result across the x-axis (Rx-axis):
- B = (3, -3)
- C = (-4, -1)
- D = (0, 2)

Resulting coordinates after (Rx-axis ∘ Ry-axis)(△BCD):
- B' = (3, -3)
- C' = (-4, -1)
- D' = (0, 2)

This is equal to applying the transformation r(180°, O) directly on triangle BCD, which gives:
- B = (-3, 3) → (3, -3)
- C = (4, 1) → (-4, -1)
- D = (0, -2) → (0, 2)

So we can conclude:
**The image of (Rx-axis ∘ Ry-axis)(△BCD) is the image r(180°, O) (△BCD).**

2. **(Ry-axis ∘ Rx-axis)(△BCD)**:
- First, reflect across the x-axis (Rx-axis):
- B = (-3, -3)
- C = (4, -1)
- D = (0, 2)
- Now reflect this result across the y-axis (Ry-axis):
- B = (3, -3)
- C = (-4, -1)
- D = (0, 2)

Resulting coordinates after (Ry-axis ∘ Rx-axis)(△BCD):
- B' = (3, -3)
- C' = (-4, -1)
- D' = (0, 2)

This too will give us the same result as applying the 180-degree rotation directly:
- B = (-3, 3) → (3, -3)
- C = (4, 1) → (-4, -1)
- D = (0, -2) → (0, 2)

So again we find:
**The image of (Ry-axis ∘ Rx-axis)(△BCD) is the image r(180°, O) (△BCD).**

### Final Answers:
1. **(Rx-axis ∘ Ry-axis)(△BCD)** → **the image r(180°, O) (△BCD).**
2. **(Ry-axis ∘ Rx-axis)(△BCD)** → **the image r(180°, O) (△BCD).**