Question

Consider ΔABC , with vertices A(0,3) , B(0,1) , and C(−4,2) , and ΔDEF , 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)
Responses

ΔABC is dilated by a scale factor of 12 and is reflected across the x-axis.
triangle upper A upper B upper C is dilated by a scale factor of Start Fraction 1 over 2 End Fraction and is reflected across the x -axis.

ΔABC is dilated by a scale factor of 14 and is reflected across the y-axis.
triangle upper A upper B upper C is dilated by a scale factor of Start Fraction 1 over 4 End Fraction and is reflected across the y -axis.

ΔABC is dilated by a scale factor of −2 and is reflected across the y-axis.
triangle upper A upper B upper C is dilated by a scale factor of negative 2 and is reflected across the y -axis.

ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.
triangle upper A upper B upper C is dilated by a scale factor of 2 and is reflected across the y -axis.
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Answer 1

To determine if triangles \( \Delta ABC \) and \( \Delta DEF \) are similar, we can analyze their sides and check if a transformation can be applied to \( \Delta ABC \) to match \( \Delta DEF \).

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

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

### Step 1: Calculating Side Lengths
1. **For \( \Delta ABC \)**:
- Side \( AB \): \( \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2 \)
- Side \( BC \): \( \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \)
- Side \( CA \): \( \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \)

2. **For \( \Delta DEF \)**:
- Side \( DE \): \( \sqrt{(0 - 0)^2 + (6 - 2)^2} = \sqrt{0 + 16} = 4 \)
- Side \( EF \): \( \sqrt{(0 - 8)^2 + (2 - 4)^2} = \sqrt{(8)^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} \)
- Side \( FD \): \( \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} \)

### Step 2: Comparing Ratios of Sides
Now we compare the ratios of corresponding sides to see if they are proportional:

- Compare \( \frac{AB}{DE} = \frac{2}{4} = \frac{1}{2} \)
- Compare \( \frac{BC}{EF} = \frac{\sqrt{17}}{\sqrt{68}} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \)
- Compare \( \frac{CA}{FD} = \frac{\sqrt{17}}{\sqrt{68}} = \frac{1}{2} \)

### Step 3: Transformations
Since all the side ratios are the same (\( \frac{1}{2} \)) and triangles share the same angles due to proportional sides, the triangles are similar.

If \( \Delta ABC \) is dilated by a factor of 2 to become \( \Delta DEF \):
- The scaling is also consistent. Thus:
- \( \Delta ABC \) can be dilated by a factor of 2 and does not require reflection to align with \( \Delta DEF \).

### Conclusion
The correct transformation is:
**ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis**.

Thus, the answer is:
- **ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.**