Three resistance each of `4Omega` are connected to form a triangle. The resistance between any two terminals is

Options

A. `12Omega`
B. `2Omega`
C. `6Omega`
D. `(8)/(3)Omega`

Answer

To find the equivalent resistance between any two terminals of a triangle formed by three resistances of 4 ohms each, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
We have three resistors, each of 4 ohms, connected in a triangular configuration. Let's label the vertices of the triangle as A, B, and C.

2. **Visualize the Circuit**:
The resistances can be represented as follows:
- Resistance R1 (between A and B) = 4 ohms
- Resistance R2 (between B and C) = 4 ohms
- Resistance R3 (between C and A) = 4 ohms

3. **Choose Two Terminals**:
We will find the equivalent resistance between terminals A and B.

4. **Analyze the Circuit**:
When a current enters terminal A, it can either go directly to terminal B through R1 or go to terminal C through R3 and then to terminal B through R2.

5. **Combine Resistors**:
The resistors R2 and R3 are in series when considering the path from A to B through C. Therefore, their equivalent resistance (R23) is:
\[ R_{23} = R2 + R3 = 4 \, \Omega + 4 \, \Omega = 8 \, \Omega \]

6. **Combine with R1**:
Now, the equivalent resistance R23 (8 ohms) is in parallel with R1 (4 ohms). The formula for the equivalent resistance (R_eq) of two resistors in parallel (R1 and R23) is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R1} + \frac{1}{R23} \]
Substituting the values:
\[ \frac{1}{R_{eq}} = \frac{1}{4} + \frac{1}{8} \]
To combine these fractions, we find a common denominator (8):
\[ \frac{1}{R_{eq}} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]

7. **Calculate R_eq**:
Taking the reciprocal gives us:
\[ R_{eq} = \frac{8}{3} \, \Omega \]

### Final Answer:
The equivalent resistance between any two terminals (A and B) is:
\[ R_{eq} = \frac{8}{3} \, \Omega \]