`Delta ABC` is similar to `Delta DEF`. The ratio of their perimeters is 4 :1 The ratio of their areas is

Options

A. ` 4 :1`
B. `16 :1`
C. `8 :1`
D. ` 8 sqrt(2) : 1`

Answer

To find the ratio of the areas of two similar triangles, we can use the relationship between the ratio of their perimeters and the ratio of their areas.

**Step-by-step solution:**

1. **Understanding the relationship between perimeters and sides:**
- When two triangles are similar, the ratio of their corresponding sides is equal to the ratio of their perimeters.
- Given that the ratio of the perimeters of triangle ABC to triangle DEF is 4:1, we can denote the ratio of their corresponding sides as:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = \frac{4}{1} \]

2. **Finding the ratio of the sides:**
- From the perimeter ratio, we conclude that the ratio of the corresponding sides of triangle ABC to triangle DEF is also 4:1.

3. **Using the ratio of sides to find the ratio of areas:**
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
- Therefore, we can calculate the ratio of the areas as follows:
\[ \text{Ratio of areas} = \left(\frac{\text{Side of ABC}}{\text{Side of DEF}}\right)^2 = \left(\frac{4}{1}\right)^2 \]

4. **Calculating the area ratio:**
- Squaring the ratio of the sides:
\[ \left(\frac{4}{1}\right)^2 = \frac{16}{1} \]

5. **Final answer:**
- Thus, the ratio of the areas of triangle ABC to triangle DEF is:
\[ 16:1 \]

**Conclusion:**
The ratio of the areas of triangle ABC to triangle DEF is 16:1.

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