The difference between the area of a square and that of an equilateral triangle on the same base is 1/4 `cm^(2)`. What is the length of side of triangle ?

Options

A. `(4 - sqrt(3))^(1//2)` cm
B. `(4 + sqrt(3))^(1//2)` cm
C. `(4 - sqrt(3))^(-1//2)` cm
D. `(4 + sqrt(3))^(-1//2)` cm

Answer

To solve the problem, we need to find the length of the side of an equilateral triangle given that the difference between the area of a square and that of the triangle on the same base is \( \frac{1}{4} \, \text{cm}^2 \).

### Step-by-Step Solution:

1. **Define Variables**:
Let the side length of the equilateral triangle be \( a \, \text{cm} \). Since the square and triangle share the same base, the side length of the square is also \( a \, \text{cm} \).

2. **Calculate the Area of the Square**:
The area of the square is given by the formula:
\[ \text{Area of square} = a^2 \]

3. **Calculate the Area of the Equilateral Triangle**:
The area of an equilateral triangle is given by the formula:
\[ \text{Area of triangle} = \frac{\sqrt{3}}{4} a^2 \]

4. **Set Up the Equation**:
According to the problem, the difference between the area of the square and the area of the triangle is \( \frac{1}{4} \, \text{cm}^2 \):
\[ a^2 - \frac{\sqrt{3}}{4} a^2 = \frac{1}{4} \]

5. **Simplify the Equation**:
Combine the terms on the left side:
\[ \left(1 - \frac{\sqrt{3}}{4}\right) a^2 = \frac{1}{4} \]
To simplify \( 1 - \frac{\sqrt{3}}{4} \), we can express \( 1 \) as \( \frac{4}{4} \):
\[ \left(\frac{4 - \sqrt{3}}{4}\right) a^2 = \frac{1}{4} \]

6. **Eliminate the Denominator**:
Multiply both sides by \( 4 \) to eliminate the fraction:
\[ (4 - \sqrt{3}) a^2 = 1 \]

7. **Solve for \( a^2 \)**:
Divide both sides by \( (4 - \sqrt{3}) \):
\[ a^2 = \frac{1}{4 - \sqrt{3}} \]

8. **Take the Square Root**:
To find \( a \), take the square root of both sides:
\[ a = \sqrt{\frac{1}{4 - \sqrt{3}}} \]

9. **Rationalize the Denominator**:
To simplify further, multiply the numerator and denominator by \( 4 + \sqrt{3} \):
\[ a = \sqrt{\frac{4 + \sqrt{3}}{(4 - \sqrt{3})(4 + \sqrt{3})}} = \sqrt{\frac{4 + \sqrt{3}}{16 - 3}} = \sqrt{\frac{4 + \sqrt{3}}{13}} \]

### Final Result:
Thus, the length of the side of the equilateral triangle is:
\[ a = \sqrt{\frac{4 + \sqrt{3}}{13}} \, \text{cm} \]