Question

Consider
A.ABC, with vertices
A (0,3),
B (0, 1), and C(-4,2), and
ADEF, with vertices D (0, 6), E (0, 2), and
F (8,4). Which transformations could be used to show that the triangles are similar? (1 point)
AABC is dilated by a scale factor of -2 and is reflected across the y-
axis.
• AABC is dilated by a scale factor of - and is reflected across the y-axis
• AABC is dilated by a scale factor of 1 and is reflected across the x-axis.
• AABC is dilated by a scale factor of 2 and is reflected across the y-axis.

Answer 1

To determine which transformation(s) could be used to show that the triangles A.ABC and ADEF are similar, we first need to analyze the coordinates of the given vertices of both triangles.

### Vertices:
- Triangle ABC:
- A(0, 3)
- B(0, 1)
- C(-4, 2)

- Triangle DEF:
- D(0, 6)
- E(0, 2)
- F(8, 4)

### Step 1: Calculate side lengths
We can find the lengths of the sides of both triangles to check for similarity.

**Triangle ABC:**
- AB = distance from A to B = |3 - 1| = 2
- BC = distance from B to C = √((0 - (-4))² + (1 - 2)²) = √(4² + (-1)²) = √(16 + 1) = √17
- AC = distance from A to C = √((0 - (-4))² + (3 - 2)²) = √(4² + 1²) = √(16 + 1) = √17

**Triangle DEF:**
- DE = distance from D to E = |6 - 2| = 4
- EF = distance from E to F = √((0 - 8)² + (2 - 4)²) = √((-8)² + (-2)²) = √(64 + 4) = √68
- DF = distance from D to F = √((0 - 8)² + (6 - 4)²) = √((-8)² + (2)²) = √(64 + 4) = √68

### Step 2: Compare the sides
From the calculations:
- Triangle ABC side lengths: 2, √17, √17
- Triangle DEF side lengths: 4, √68, √68

### Step 3: Scale factor
To find a scale factor from triangle ABC to DEF:
- The ratio of the corresponding sides AB and DE is 4/2 = 2.
- The ratio of sides BC (√17) and EF (√68) can be simplified:
- √68 / √17 = √(68/17) = √4 = 2.
- The same holds for AC (√17) and DF (√68) as well (same ratio).

This shows that a dilation by a scale factor of 2 maintains the ratio of the corresponding sides.

### Step 4: Reflection and transformations
Reflecting across the y-axis or x-axis and performing a dilation maintains the similarity.

### Potential Transformation:

1. **AABC is dilated by a scale factor of 2 and is reflected across the y-axis.**
- Dilation by 2 scales all dimensions but also ensure that we have the same arrangement to match the vertices correctly.

The other options either suggest a negative scale transformation or incorrect reflection that don’t result in similarity, based on our calculations.

### Conclusion:
The valid transformation showing that the triangles are similar is:

**AABC is dilated by a scale factor of 2 and is reflected across the y-axis.**