Three resistor, each equal to 32, are connected so as to form a triangle. The equivalent resistance between any two vertices of the triangle is :

Question

Accepted Answer

To determine the equivalent resistance between any two vertices of a triangle formed by three resistors, each with a resistance of 3 Ω, we need to analyze the circuit configuration.

### Explanation for Option 2:

1. **Circuit Configuration**: The three resistors are connected in a triangle (or delta) configuration. This means each resistor is connected between two vertices of the triangle.

2. **Equivalent Resistance Calculation**: When calculating the equivalent resistance between any two vertices in a delta configuration, we consider the two resistors directly connected between the vertices and the third resistor, which is connected across the opposite side.

- Let's consider vertices A, B, and C, with resistors R1, R2, and R3 each equal to 3 Ω.
   - To find the equivalent resistance between vertices A and B, we consider R1 (between A and B), R2 (between B and C), and R3 (between C and A).

3. **Using Delta to Star Transformation**: In a delta configuration, the equivalent resistance between any two vertices (say A and B) can be calculated using the formula:
   $$
   R_{AB} = \frac{R1 \times R2 + R2 \times R3 + R3 \times R1}{R3}
   $$
   Substituting the values:
   $$
   R_{AB} = \frac{3 \times 3 + 3 \times 3 + 3 \times 3}{3} = \frac{9 + 9 + 9}{3} = \frac{27}{3} = 9 \ \Omega
   $$

4. **Conclusion**: The equivalent resistance between any two vertices of the triangle is 9 Ω, which matches Option 2.

### Explanation for Incorrect Options:

- **Option 1 ($1 \ \Omega$)**: This value is too low for the given configuration. The resistors are not in series or parallel in a way that would result in such a low equivalent resistance.

- **Option 3 ($3 \ \Omega$)**: This value represents the resistance of a single resistor, not the equivalent resistance between two vertices in a delta configuration.

- **Option 4 ($9 \ \Omega$)**: This value is incorrect as it does not match the calculated equivalent resistance using the delta configuration formula.

In conclusion, Option 2 is the correct answer as it accurately represents the equivalent resistance between any two vertices of the triangle formed by three 3 Ω resistors.