If an equilateral triangle is made of a uniform wire is resistance R, the equivalent resistance between the ends of sides is

Options

A. `(2R)/(3)`
B. `(R )/(3)`
C. `(2R)/(9)`
D. `(9R)/(2)`

Explanation

To find the equivalent resistance between the ends of the sides of an equilateral triangle made of a uniform wire with total resistance \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure**:
- An equilateral triangle consists of three equal sides. When a uniform wire of resistance \( R \) is shaped into an equilateral triangle, the total resistance \( R \) is distributed equally among the three sides.

2. **Calculate Resistance of Each Side**:
- Since the total resistance \( R \) is divided equally among the three sides, the resistance of each side (let's call it \( R_s \)) is:
\[ R_s = \frac{R}{3} \]

3. **Identify the Points of Interest**:
- Let’s denote the vertices of the triangle as points \( P \), \( Q \), and \( R \). We want to find the equivalent resistance between points \( P \) and \( Q \).

4. **Analyze the Current Flow**:
- When a current enters at point \( P \), it can either flow directly to point \( Q \) through one side or split and flow through the other two sides before reaching point \( Q \).

5. **Determine the Configuration**:
- The two sides \( PR \) and \( QR \) are in parallel when considering the current flowing from \( P \) to \( Q \). The resistance of side \( PR \) is \( R_s \) and the resistance of side \( QR \) is also \( R_s \).

6. **Calculate the Equivalent Resistance of the Parallel Combination**:
- The equivalent resistance \( R_{PQ} \) for two resistances \( R_s \) in parallel is given by:
\[ \frac{1}{R_{PQ}} = \frac{1}{R_s} + \frac{1}{R_s} = \frac{2}{R_s} \]
- Substituting \( R_s = \frac{R}{3} \):
\[ \frac{1}{R_{PQ}} = \frac{2}{\frac{R}{3}} = \frac{6}{R} \]
- Therefore, the equivalent resistance \( R_{PQ} \) is:
\[ R_{PQ} = \frac{R}{6} \]

7. **Add the Resistance of the Direct Path**:
- The direct path from \( P \) to \( Q \) through one side of the triangle also has resistance \( R_s \) which is \( \frac{R}{3} \).

8. **Combine the Resistances**:
- The equivalent resistance \( R_{eq} \) between points \( P \) and \( Q \) is the sum of the equivalent resistance of the parallel combination and the resistance of the direct path:
\[ R_{eq} = R_{PQ} + R_s = \frac{R}{6} + \frac{R}{3} \]
- Converting \( \frac{R}{3} \) to sixths:
\[ R_{eq} = \frac{R}{6} + \frac{2R}{6} = \frac{3R}{6} = \frac{R}{2} \]

### Final Answer:
The equivalent resistance between the ends of the sides of the equilateral triangle is:
\[ R_{eq} = \frac{R}{2} \]