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)

Answer 1

To determine whether triangles \( \Delta ABC \) and \( \Delta DEF \) are similar through transformations, we need to analyze the shapes and sides of both triangles.

1. **Identify the vertices of both triangles**:
- For \( \Delta ABC \):
- A(0, 3)
- B(0, 1)
- C(-4, 2)

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

2. **Calculate the lengths of the sides of each triangle**:
- For \( \Delta ABC \):
- AB = Distance between A(0,3) and B(0,1) = \( |3 - 1| = 2 \)
- BC = Distance between B(0,1) and C(-4,2) = \( \sqrt{(-4 - 0)^2 + (2 - 1)^2} = \sqrt{16 + 1} = \sqrt{17} \)
- CA = Distance between C(-4,2) and A(0,3) = \( \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{16 + 1} = \sqrt{17} \)

- For \( \Delta DEF \):
- DE = Distance between D(0,6) and E(0,2) = \( |6 - 2| = 4 \)
- EF = Distance between E(0,2) and F(8,4) = \( \sqrt{(8 - 0)^2 + (4 - 2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)
- FD = Distance between F(8,4) and D(0,6) = \( \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \)

3. **Comparing the side lengths**:
- For \( \Delta ABC \):
- Lengths: 2, \( \sqrt{17} \), \( \sqrt{17} \)
- For \( \Delta DEF \):
- Lengths: 4, \( 2\sqrt{17} \), \( 2\sqrt{17} \)

4. **Determine the ratio of corresponding sides**:
- The corresponding sides of the triangles can be expressed as follows:
- \( \frac{AB}{DE} = \frac{2}{4} = \frac{1}{2} \)
- \( \frac{BC}{EF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \)
- \( \frac{CA}{FD} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \)

Since the side lengths of both triangles follow the same ratio \( \frac{1}{2} \), this means the two triangles are similar by the Side-Side-Side (SSS) similarity criterion.

5. **Transformations**:
To show that \( \Delta ABC \) is similar to \( \Delta DEF \), the following transformations could be used:

- **Scaling**: A transformation that enlarges \( \Delta ABC \) by a factor of 2, which will make the lengths of the sides of \( \Delta ABC \) match those of \( \Delta DEF \).

- **Translation or rotation** (if necessary): If the orientation is not aligned, you could also translate or rotate \( \Delta ABC \) to match the position and orientation of \( \Delta DEF \).

In summary, the triangles can be made similar through scaling (enlargement) and possibly translation or rotation.