The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is `160` cm, then what can you say about breadth ?

Options

A. breadth `=20`
B. breadth `le20`
C. breadth `ge20`
D. breadth `ne20`

Answer

To solve the problem step by step, we start by defining the variables and using the information provided in the question.

### Step-by-step Solution:

1. **Define the Variables**:
- Let the breadth of the rectangle be \( b = x \).
- According to the problem, the length \( l \) is three times the breadth. Therefore, we have:
\[ l = 3x \]

2. **Write the Perimeter Formula**:
- The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) \]
- Substituting the expressions for length and breadth, we get:
\[ P = 2(3x + x) = 2(4x) = 8x \]

3. **Set the Perimeter Equal to the Given Value**:
- We know from the problem that the minimum perimeter is \( 160 \) cm. Therefore, we can set up the equation:
\[ 8x = 160 \]

4. **Solve for \( x \)**:
- To find \( x \), divide both sides of the equation by \( 8 \):
\[ x = \frac{160}{8} = 20 \]

5. **Conclusion**:
- Since \( x \) represents the breadth of the rectangle, we conclude that:
\[ \text{Breadth } (b) = 20 \text{ cm} \]

6. **Determine the Minimum Value**:
- Since the problem states that this is the minimum perimeter, we can conclude that the breadth must be at least \( 20 \) cm. Therefore, we can express this as:
\[ b \geq 20 \text{ cm} \]

### Final Answer:
The breadth of the rectangle is \( 20 \) cm, and it must be greater than or equal to \( 20 \) cm.