`{:("Column A","ABC and DEF are right triangles. Each side of"DeltaABC" is twice the length of the corresponding side of "DeltaDBF, "Column B"),(("The area of" DeltaDEF)/("The area of" Delta ABC),,1//2):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the areas of two right triangles, ABC and DEF, and determine the ratio of their areas.

### Step-by-Step Solution:

1. **Understand the Problem**: We have two right triangles, ABC and DEF. Each side of triangle ABC is twice the length of the corresponding side of triangle DEF.

2. **Define the Sides**:
- Let the sides of triangle DEF be:
- DE = x
- EF = y
- DF = z
- Then the sides of triangle ABC will be:
- AB = 2x
- BC = 2y
- AC = 2z

3. **Calculate the Area of Triangle DEF**:
- The area of triangle DEF can be calculated using the formula for the area of a triangle:
\[ \text{Area}_{DEF} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, we can take DE as the base and EF as the height:
\[ \text{Area}_{DEF} = \frac{1}{2} \times DE \times EF = \frac{1}{2} \times x \times y \]

4. **Calculate the Area of Triangle ABC**:
- Similarly, for triangle ABC:
\[ \text{Area}_{ABC} = \frac{1}{2} \times AB \times BC \]
- Substituting the values of AB and BC:
\[ \text{Area}_{ABC} = \frac{1}{2} \times (2x) \times (2y) = \frac{1}{2} \times 4xy = 2xy \]

5. **Find the Ratio of the Areas**:
- Now we need to find the ratio of the area of triangle DEF to the area of triangle ABC:
\[ \frac{\text{Area}_{DEF}}{\text{Area}_{ABC}} = \frac{\frac{1}{2}xy}{2xy} \]
- Simplifying this expression:
\[ \frac{\text{Area}_{DEF}}{\text{Area}_{ABC}} = \frac{1}{2} \div 2 = \frac{1}{4} \]

6. **Compare with Column B**:
- Column A gives us the ratio of the areas, which we found to be \(\frac{1}{4}\).
- Column B is given as \(\frac{1}{2}\).

7. **Conclusion**:
- Since \(\frac{1}{4} < \frac{1}{2}\), we conclude that Column B is larger.

### Final Answer:
The correct option is **2** (Column B is larger).