If area of an equilateral triangle is a and height b, then value of `(b^(2))/(a)` is

Options

A. 3
B. `(1)/(3)`
C. `sqrt(3)`
D. `(1)/(sqrt(3))`

Answer

To solve the problem, we need to find the value of \(\frac{b^2}{a}\) where \(a\) is the area of an equilateral triangle and \(b\) is its height.

### Step-by-Step Solution:

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

2. **Substitute the height into the expression for \(b^2\):**
- Since \(b\) is the height, we have:
\[ b = h = \frac{\sqrt{3}}{2} s \]
- Now, calculate \(b^2\):
\[ b^2 = \left(\frac{\sqrt{3}}{2} s\right)^2 = \frac{3}{4} s^2 \]

3. **Substitute the area into the expression for \(a\):**
- We already have the area \(a\):
\[ a = A = \frac{\sqrt{3}}{4} s^2 \]

4. **Now, substitute \(b^2\) and \(a\) into the expression \(\frac{b^2}{a}\):**
\[ \frac{b^2}{a} = \frac{\frac{3}{4} s^2}{\frac{\sqrt{3}}{4} s^2} \]

5. **Simplify the expression:**
- The \(s^2\) terms cancel out:
\[ \frac{b^2}{a} = \frac{3/4}{\sqrt{3}/4} = \frac{3}{\sqrt{3}} \]
- Simplifying further:
\[ \frac{3}{\sqrt{3}} = \sqrt{3} \]

6. **Final Result:**
- Therefore, the value of \(\frac{b^2}{a}\) is:
\[ \frac{b^2}{a} = \sqrt{3} \]

### Conclusion:
The final answer is \(\sqrt{3}\).