The area of a rectangle is `60 cm^(2)` and its perimeter is 34 cm, then the length of the diagonal is

Options

A. 17 cm
B. 11 cm
C. 15 cm
D. 13 cm

Answer

To find the length of the diagonal of a rectangle given its area and perimeter, we can follow these steps:

### Step 1: Set up the equations
We know that:
- The area \( A \) of the rectangle is given by the formula:
\[ A = L \times B \]
where \( L \) is the length and \( B \) is the breadth. Given that the area is \( 60 \, \text{cm}^2 \), we have:
\[ L \times B = 60 \quad \text{(1)} \]

- The perimeter \( P \) of the rectangle is given by the formula:
\[ P = 2(L + B) \]
Given that the perimeter is \( 34 \, \text{cm} \), we have:
\[ 2(L + B) = 34 \quad \Rightarrow \quad L + B = 17 \quad \text{(2)} \]

### Step 2: Solve the equations
From equation (2), we can express \( B \) in terms of \( L \):
\[ B = 17 - L \quad \text{(3)} \]

Now, substitute equation (3) into equation (1):
\[ L \times (17 - L) = 60 \]
Expanding this gives:
\[ 17L - L^2 = 60 \]
Rearranging the equation:
\[ L^2 - 17L + 60 = 0 \]

### Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ L^2 - 17L + 60 = 0 \]
This can be factored as:
\[ (L - 12)(L - 5) = 0 \]

### Step 4: Find the values of \( L \) and \( B \)
Setting each factor to zero gives us:
\[ L - 12 = 0 \quad \Rightarrow \quad L = 12 \]
\[ L - 5 = 0 \quad \Rightarrow \quad L = 5 \]

Thus, we have two possible pairs for \( (L, B) \):
1. \( L = 12 \) and \( B = 5 \)
2. \( L = 5 \) and \( B = 12 \)

### Step 5: Calculate the length of the diagonal
The length of the diagonal \( D \) of a rectangle can be calculated using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]

Using \( L = 12 \) and \( B = 5 \):
\[ D = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{cm} \]

### Final Answer
The length of the diagonal is \( 13 \, \text{cm} \).

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