If the numerical values of the height and the area of an equilateral triangle be same, then the length of each side of the triangle is

Options

A. 2 units
B. 4 units
C. 5 units
D. 8 units

Answer

To solve the problem, we need to find the length of each side of an equilateral triangle given that the numerical values of its height and area are the same.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The height (h) of an equilateral triangle with side length \( s \) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
- The area (A) of an equilateral triangle with side length \( s \) is given by:
\[ A = \frac{\sqrt{3}}{4} s^2 \]

2. **Set the height equal to the area**:
According to the problem, the numerical values of the height and the area are the same. Therefore, we can set the two equations equal to each other:
\[ \frac{\sqrt{3}}{2} s = \frac{\sqrt{3}}{4} s^2 \]

3. **Eliminate the common factor**:
Since both sides have a common factor of \( \sqrt{3} \), we can divide both sides by \( \sqrt{3} \) (assuming \( s \neq 0 \)):
\[ \frac{1}{2} s = \frac{1}{4} s^2 \]

4. **Multiply through by 4 to eliminate the fractions**:
\[ 2s = s^2 \]

5. **Rearrange the equation**:
\[ s^2 - 2s = 0 \]

6. **Factor the equation**:
\[ s(s - 2) = 0 \]

7. **Solve for \( s \)**:
This gives us two solutions:
\[ s = 0 \quad \text{or} \quad s = 2 \]
Since \( s = 0 \) does not represent a valid triangle, we take:
\[ s = 2 \]

### Final Answer:
The length of each side of the triangle is \( 2 \).

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