`A B C` is an equilateral triangle of side `4c mdot` If `R ,r and h` are the circumradius, inradius, and altitude, respectively, then `(R+r)/h` is equal to

Options

A. 4
B. 2
C. 1
D. 3

Explanation

To solve the problem, we need to find the value of \((R + r) / h\) for an equilateral triangle \(ABC\) with a side length of \(4 \, \text{cm}\).

### Step-by-Step Solution:

1. **Identify the Side Length**:
The side length of the equilateral triangle \(ABC\) is given as \(a = 4 \, \text{cm}\).

2. **Calculate the Circumradius \(R\)**:
For an equilateral triangle, the circumradius \(R\) can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]
Substituting the value of \(a\):
\[ R = \frac{4}{\sqrt{3}} \approx 2.309 \, \text{cm} \]

3. **Calculate the Inradius \(r\)**:
The inradius \(r\) for an equilateral triangle can be calculated using the formula:
\[ r = \frac{a}{2\sqrt{3}} \]
Substituting the value of \(a\):
\[ r = \frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}} \approx 1.155 \, \text{cm} \]

4. **Calculate the Altitude \(h\)**:
The altitude \(h\) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Substituting the value of \(a\):
\[ h = \frac{\sqrt{3}}{2} \cdot 4 = 2\sqrt{3} \approx 3.464 \, \text{cm} \]

5. **Calculate \((R + r)\)**:
Now, we can calculate \(R + r\):
\[ R + r = \frac{4}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{4 + 2}{\sqrt{3}} = \frac{6}{\sqrt{3}} = 2\sqrt{3} \]

6. **Calculate \((R + r) / h\)**:
Finally, we find \((R + r) / h\):
\[ \frac{R + r}{h} = \frac{2\sqrt{3}}{2\sqrt{3}} = 1 \]

### Final Answer:
\[ \frac{R + r}{h} = 1 \]