A wire of resistance `18Omega` is divided into three equal parts. These parts are connected as sides of a triangle , the equivalent resistance of any two corners of the triangle will be

Options

A. `18Omega`
B. `9 Omega`
C. `6Omega`
D. `4Omega`

Answer

To solve the problem of finding the equivalent resistance between any two corners of a triangle formed by three equal parts of a wire with a total resistance of 18 ohms, we can follow these steps:

### Step 1: Determine the resistance of each part
The total resistance of the wire is given as 18 ohms. When the wire is divided into three equal parts, the resistance of each part can be calculated as follows:

\[ R_{\text{each}} = \frac{R_{\text{total}}}{3} = \frac{18 \, \Omega}{3} = 6 \, \Omega \]

### Step 2: Visualize the triangle configuration
Now, we have three resistors, each with a resistance of 6 ohms, connected in the shape of a triangle. We can label the corners of the triangle as A, B, and C, and the resistors as follows:
- Resistor between A and B: \( R_{AB} = 6 \, \Omega \)
- Resistor between B and C: \( R_{BC} = 6 \, \Omega \)
- Resistor between C and A: \( R_{CA} = 6 \, \Omega \)

### Step 3: Find the equivalent resistance between two corners
To find the equivalent resistance between any two corners, say A and B, we need to consider the resistors connected to these corners. The resistors \( R_{AB} \) and \( R_{CA} \) are in series, and \( R_{BC} \) is in parallel with the series combination of \( R_{AB} \) and \( R_{CA} \).

1. Calculate the total resistance of the series combination \( R_{AB} + R_{CA} \):

\[ R_{AB} + R_{CA} = 6 \, \Omega + 6 \, \Omega = 12 \, \Omega \]

2. Now, we have \( R_{BC} \) in parallel with the total resistance from step 1:

\[ R_{\text{eq}} = \frac{R_{12} \cdot R_{BC}}{R_{12} + R_{BC}} = \frac{12 \, \Omega \cdot 6 \, \Omega}{12 \, \Omega + 6 \, \Omega} \]

### Step 4: Calculate the equivalent resistance
Now, substituting the values into the formula:

\[ R_{\text{eq}} = \frac{12 \cdot 6}{12 + 6} = \frac{72}{18} = 4 \, \Omega \]

### Conclusion
Thus, the equivalent resistance between any two corners of the triangle is:

\[ \boxed{4 \, \Omega} \]