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 \(4 \Omega\) each, we can follow these steps:

### Step 1: Identify the Configuration
We have three resistors, each of resistance \(R = 4 \Omega\), connected in a triangular configuration. Let's label the vertices of the triangle as A, B, and C, with resistors as follows:
- \(R_{AB} = 4 \Omega\) (between A and B)
- \(R_{BC} = 4 \Omega\) (between B and C)
- \(R_{CA} = 4 \Omega\) (between C and A)

### Step 2: Choose Two Terminals
We will calculate the equivalent resistance between terminals A and B.

### Step 3: Analyze the Circuit
When looking at the resistors connected to terminals A and B:
- The resistor \(R_{AB}\) is directly between A and B.
- The resistors \(R_{AC}\) and \(R_{BC}\) are connected in series between A and C and B and C, respectively.

### Step 4: Calculate Total Resistance
1. **Calculate the resistance of the two resistors \(R_{AC}\) and \(R_{BC}\) in series:**
\[ R_{AC} + R_{BC} = 4 \Omega + 4 \Omega = 8 \Omega \]

2. **Now, we have \(R_{AB}\) (4 Ω) in parallel with the total resistance of \(R_{AC} + R_{BC}\) (8 Ω):**
\[ R_{AB} \text{ (4 Ω)} \text{ is in parallel with } R_{AC} + R_{BC} \text{ (8 Ω)} \]

3. **Using the formula for resistors in parallel:**
\[ R_{AB} = \frac{R_1 \cdot R_2}{R_1 + R_2} \]
where \(R_1 = 4 \Omega\) and \(R_2 = 8 \Omega\):
\[ R_{AB} = \frac{4 \cdot 8}{4 + 8} = \frac{32}{12} = \frac{8}{3} \Omega \]

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

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