A letter `A` is constructed as a uniform wire of resistance 1 ohm/cm. The sides of the letter are 20 cm long and the cross piece in the middle is 10 cm long while the vertex angle is `60^(@)` the resistance of the letter between the two ends of the legs is

Options

A. `(40)/(3)Omega`
B. `(80)/(3)Omega`
C. `40Omega`
D. `10Omega`

Explanation

To solve the problem of finding the resistance of the letter 'A' constructed from a uniform wire of resistance 1 ohm/cm, we will break down the problem step by step.

### Step 1: Understand the Structure of the Letter 'A'
The letter 'A' consists of:
- Two legs, each 20 cm long.
- A cross piece in the middle that is 10 cm long.
- The vertex angle at the top is 60 degrees.

### Step 2: Calculate the Resistance of Each Segment
Given that the resistance of the wire is 1 ohm/cm, we can calculate the resistance of each segment:
- **Legs (2 segments of 20 cm each)**:
\[ R_{\text{leg}} = 20 \, \text{cm} \times 1 \, \text{ohm/cm} = 20 \, \text{ohms} \]
Since there are two legs, the total resistance for both legs in series is:
\[ R_{\text{legs}} = R_{\text{leg}} + R_{\text{leg}} = 20 \, \text{ohms} + 20 \, \text{ohms} = 40 \, \text{ohms} \]

- **Cross Piece (1 segment of 10 cm)**:
\[ R_{\text{cross}} = 10 \, \text{cm} \times 1 \, \text{ohm/cm} = 10 \, \text{ohms} \]

### Step 3: Analyze the Circuit Configuration
The configuration of the letter 'A' can be viewed as follows:
- The two legs are connected in parallel with the cross piece.
- The resistance of the cross piece is in series with the equivalent resistance of the two legs.

### Step 4: Calculate the Equivalent Resistance of the Legs
Since the two legs are in parallel, we can calculate their equivalent resistance \( R_{\text{eq, legs}} \) using the formula for resistors in parallel:
\[ \frac{1}{R_{\text{eq, legs}}} = \frac{1}{R_{\text{leg}}} + \frac{1}{R_{\text{leg}}} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]
Thus,
\[ R_{\text{eq, legs}} = 10 \, \text{ohms} \]

### Step 5: Combine the Equivalent Resistance with the Cross Piece
Now, we need to add the resistance of the cross piece to the equivalent resistance of the legs:
\[ R_{\text{total}} = R_{\text{eq, legs}} + R_{\text{cross}} = 10 \, \text{ohms} + 10 \, \text{ohms} = 20 \, \text{ohms} \]

### Final Answer
The resistance of the letter 'A' between the two ends of the legs is:
\[ \boxed{20 \, \text{ohms}} \]