If every side of an equilateral triangle is doubled, the area of the new triangle is K times the area of the old one. K is equal to

Options

A. `sqrt(2)`
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the value of K, which represents how many times the area of the new equilateral triangle (with doubled sides) is compared to the area of the original triangle.

### Step-by-Step Solution:

1. **Define the side of the original triangle**:
Let the side length of the original equilateral triangle be \( a \).

2. **Calculate the area of the original triangle**:
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

3. **Define the side of the new triangle**:
If every side of the triangle is doubled, the side length of the new triangle becomes \( 2a \).

4. **Calculate the area of the new triangle**:
Using the same area formula for the new triangle with side length \( 2a \):
\[ A_1 = \frac{\sqrt{3}}{4} (2a)^2 \]
Simplifying this:
\[ A_1 = \frac{\sqrt{3}}{4} \cdot 4a^2 = \sqrt{3} a^2 \]

5. **Relate the areas of the new and old triangles**:
Now, we can express the area of the new triangle in terms of the area of the old triangle:
\[ A_1 = \sqrt{3} a^2 \]
Since the area of the original triangle \( A = \frac{\sqrt{3}}{4} a^2 \), we can find the ratio:
\[ K = \frac{A_1}{A} = \frac{\sqrt{3} a^2}{\frac{\sqrt{3}}{4} a^2} \]
Simplifying this gives:
\[ K = \frac{\sqrt{3} a^2 \cdot 4}{\sqrt{3} a^2} = 4 \]

6. **Conclusion**:
Therefore, the value of \( K \) is 4.

### Final Answer:
K is equal to 4.

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