The perimeter of rectangle is equal to the perimeter of a right angle triangle of height 12 cm. If base of the triangle is equal to the breadth of the rectangle .What is the length of the rectangle ?

Options

A. 18 cms
B. 24 cms
C. 22 cms
D. Data inadequate

Answer

To solve the problem, we need to find the length of the rectangle given that the perimeter of the rectangle is equal to the perimeter of a right-angled triangle with a height of 12 cm, and the base of the triangle is equal to the breadth of the rectangle.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the breadth of the rectangle be \( b \).
- Let the length of the rectangle be \( l \).
- The height of the triangle is given as \( h = 12 \) cm.
- The base of the triangle, which is equal to the breadth of the rectangle, is \( b \).

2. **Perimeter of the Rectangle**:
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P_{\text{rectangle}} = 2(l + b) \]

3. **Perimeter of the Right-Angled Triangle**:
The perimeter \( P \) of a right-angled triangle can be calculated as:
\[ P_{\text{triangle}} = \text{base} + \text{height} + \text{hypotenuse} \]
Here, the base is \( b \), and the height is \( h = 12 \) cm. We need to find the hypotenuse, which we can denote as \( c \). Using the Pythagorean theorem:
\[ c = \sqrt{b^2 + h^2} = \sqrt{b^2 + 12^2} = \sqrt{b^2 + 144} \]
Therefore, the perimeter of the triangle becomes:
\[ P_{\text{triangle}} = b + 12 + \sqrt{b^2 + 144} \]

4. **Set the Perimeters Equal**:
According to the problem, the perimeters of the rectangle and the triangle are equal:
\[ 2(l + b) = b + 12 + \sqrt{b^2 + 144} \]

5. **Simplify the Equation**:
Rearranging the equation:
\[ 2l + 2b = b + 12 + \sqrt{b^2 + 144} \]
Subtract \( b \) from both sides:
\[ 2l + b = 12 + \sqrt{b^2 + 144} \]
Now, isolate \( 2l \):
\[ 2l = 12 + \sqrt{b^2 + 144} - b \]
Finally, divide by 2 to find \( l \):
\[ l = \frac{12 + \sqrt{b^2 + 144} - b}{2} \]

6. **Conclusion**:
We have expressed the length \( l \) in terms of the breadth \( b \). However, without a specific value for \( b \), we cannot determine a numerical value for \( l \). Thus, the data provided is inadequate to find a specific length.