`Delta ABC` is similar to `Delta DEF`. If the ratio of similar sides is `k : 1`, the ratio of their areas is

Options

A. `k^(2) :1`
B. `2k:1`
C. `(k^(2))/(2) :1`
D. `2k^(2) :1`

Answer

To find the ratio of the areas of two similar triangles, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Similar Triangles**:
Given that triangle ABC is similar to triangle DEF, we know that corresponding angles are equal and the sides are in proportion.

2. **Set Up the Ratio of Sides**:
The ratio of the lengths of corresponding sides of the triangles is given as \( k : 1 \). This means if one side of triangle ABC is \( kx \), the corresponding side of triangle DEF is \( x \).

3. **Use the Area Formula**:
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For both triangles, we can express their areas in terms of their sides and the sine of the included angle.

4. **Express Areas in Terms of Sides**:
Let’s denote the area of triangle ABC as \( \Delta_1 \) and the area of triangle DEF as \( \Delta_2 \). Using the sides and the sine of the included angle (which is the same for both triangles since they are similar), we have:
\[ \Delta_1 = \frac{1}{2} \times (k \cdot \text{base}) \times (k \cdot \text{height}) \times \sin(\theta) \]
\[ \Delta_2 = \frac{1}{2} \times \text{base} \times \text{height} \times \sin(\theta) \]

5. **Simplify the Area Ratios**:
Since the sine of the angle is the same for both triangles, it cancels out when we take the ratio of their areas:
\[ \frac{\Delta_1}{\Delta_2} = \frac{\frac{1}{2} \times (k \cdot \text{base}) \times (k \cdot \text{height}) \times \sin(\theta)}{\frac{1}{2} \times \text{base} \times \text{height} \times \sin(\theta)} \]
This simplifies to:
\[ \frac{\Delta_1}{\Delta_2} = \frac{k^2 \cdot \text{base} \cdot \text{height}}{\text{base} \cdot \text{height}} = k^2 \]

6. **Conclusion**:
Therefore, the ratio of the areas of triangles ABC and DEF is:
\[ \text{Ratio of Areas} = k^2 : 1 \]

### Final Answer:
The ratio of their areas is \( k^2 : 1 \).