State whether the following statements are true or false. Justify your answer
`triangle`ABC with vertices A (-2,0),B (2,0) and C (0,2) is similar to `triangle`DEF with vertices D (-4,0) ,E (4,0) and F (0,4).

Explanation

To determine whether triangle ABC with vertices A (-2,0), B (2,0), and C (0,2) is similar to triangle DEF with vertices D (-4,0), E (4,0), and F (0,4), we will calculate the lengths of the sides of both triangles and compare their ratios.

### Step 1: Calculate the lengths of the sides of triangle ABC

1. **Length of AB**:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - (-2))^2 + (0 - 0)^2} = \sqrt{(2 + 2)^2} = \sqrt{4^2} = 4 \]

2. **Length of BC**:
\[ BC = \sqrt{(0 - 2)^2 + (2 - 0)^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

3. **Length of AC**:
\[ AC = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 2: Calculate the lengths of the sides of triangle DEF

1. **Length of DE**:
\[ DE = \sqrt{(4 - (-4))^2 + (0 - 0)^2} = \sqrt{(4 + 4)^2} = \sqrt{8^2} = 8 \]

2. **Length of EF**:
\[ EF = \sqrt{(0 - 4)^2 + (4 - 0)^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

3. **Length of DF**:
\[ DF = \sqrt{(0 - (-4))^2 + (4 - 0)^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

### Step 3: Compare the ratios of the corresponding sides

1. **Ratio of AB to DE**:
\[ \frac{AB}{DE} = \frac{4}{8} = \frac{1}{2} \]

2. **Ratio of BC to EF**:
\[ \frac{BC}{EF} = \frac{2\sqrt{2}}{4\sqrt{2}} = \frac{1}{2} \]

3. **Ratio of AC to DF**:
\[ \frac{AC}{DF} = \frac{2\sqrt{2}}{4\sqrt{2}} = \frac{1}{2} \]

### Conclusion

Since the ratios of the corresponding sides of triangles ABC and DEF are equal (all equal to \(\frac{1}{2}\)), we can conclude that triangle ABC is similar to triangle DEF.

Thus, the statement is **True**.