Question

Triangle ABC has the points A(1,1), B(7,2) and C(5,7). what composition transformation will result in triangle A"B"C" being at A" (2,-1), B" (-4,0) and C" (-2,5)?
A. a reflection across the x-axis and a move to the left by 3 and down by 2
B. a reflection across the y-axis and a move to the right by 3 down by 2
C. a rotation of -270 degrees and a move right by 3 and up by 2
D. a reflection across the y-axis and a move to the left by 5 and up by 3

Answer 1

To find the correct composition transformation from triangle ABC to triangle A"B"C", we need to analyze the transformations that take the vertices A, B, and C to A", B", and C".

The vertices of triangle ABC are:
- A(1, 1)
- B(7, 2)
- C(5, 7)

The vertices of triangle A"B"C" are:
- A"(2, -1)
- B"(-4, 0)
- C"(-2, 5)

### Step 1: Analyze the transformation

Let's perform two transformations: a reflection and a translation.

#### Reflection Across the Y-Axis
Reflecting points across the y-axis changes the x-coordinates of the points.
- Reflecting A(1, 1) across the y-axis gives A'(-1, 1).
- Reflecting B(7, 2) gives B'(-7, 2).
- Reflecting C(5, 7) gives C'(-5, 7).

#### Find the Translation Needed
Now, we need to find a translation that takes the points A'(-1, 1), B'(-7, 2), and C'(-5, 7) to points A"(2, -1), B"(-4, 0), and C"(-2, 5).

Calculating the translation vector (change in x, change in y):
1. For A'(-1, 1) to A"(2, -1):
- Change in x: 2 - (-1) = 3
- Change in y: -1 - 1 = -2
- Translation vector = (3, -2)

2. For B'(-7, 2) to B"(-4, 0):
- Change in x: -4 - (-7) = 3
- Change in y: 0 - 2 = -2
- Translation vector = (3, -2)

3. For C'(-5, 7) to C"(-2, 5):
- Change in x: -2 - (-5) = 3
- Change in y: 5 - 7 = -2
- Translation vector = (3, -2)

### Step 2: Combine the Results

Thus, we have determined the transformations needed:
1. A reflection across the y-axis.
2. A translation of (3, -2) which is to the right by 3 and down by 2.

### Conclusion

Based on the transformations we analyzed, the answer is:
**B. a reflection across the y-axis and a move to the right by 3 down by 2**.