A circular wire of length 168 cm is cut and bent in the form of an rectangle whose sides are in the ratio of 5:7. What is the length (in cm) of the diagonal of the rectangle ?

Options

A. `sqrt(4127)`
B. `sqrt(3137)`
C. `sqrt(1813)`
D. `sqrt(3626)`

Answer

To find the length of the diagonal of the rectangle formed by bending a circular wire of length 168 cm, we can follow these steps:

### Step 1: Understand the problem
We have a circular wire of length 168 cm that is bent to form a rectangle. The sides of the rectangle are in the ratio of 5:7.

### Step 2: Set up the dimensions of the rectangle
Let the length of the rectangle be \(5x\) and the breadth be \(7x\).

### Step 3: Write the equation for the perimeter
The perimeter \(P\) of a rectangle is given by the formula:
\[ P = 2(\text{length} + \text{breadth}) = 2(5x + 7x) = 2(12x) = 24x \]
Since the perimeter is equal to the length of the wire, we have:
\[ 24x = 168 \]

### Step 4: Solve for \(x\)
To find \(x\), we divide both sides of the equation by 24:
\[ x = \frac{168}{24} = 7 \]

### Step 5: Calculate the dimensions of the rectangle
Now that we have \(x\), we can find the length and breadth:
- Length = \(5x = 5 \times 7 = 35 \, \text{cm}\)
- Breadth = \(7x = 7 \times 7 = 49 \, \text{cm}\)

### Step 6: Use the Pythagorean theorem to find the diagonal
The diagonal \(d\) of a rectangle can be calculated using the Pythagorean theorem:
\[ d = \sqrt{(\text{length})^2 + (\text{breadth})^2} \]
Substituting the values we found:
\[ d = \sqrt{(35)^2 + (49)^2} \]

### Step 7: Calculate the squares
Calculating the squares:
\[ 35^2 = 1225 \]
\[ 49^2 = 2401 \]

### Step 8: Add the squares
Now, add the two results:
\[ d = \sqrt{1225 + 2401} = \sqrt{3626} \]

### Step 9: Final result
Thus, the length of the diagonal of the rectangle is:
\[ d = \sqrt{3626} \, \text{cm} \]