The perimeter and the breadth of a rectangle are 82 cm. and 20 cm. respectively. Calculate the length of its diagonal (in cm).

Options

A. 58
B. 21
C. 42
D. 29

Answer

To find the length of the diagonal of the rectangle, we will follow these steps:

### Step 1: Understand the formula for the perimeter of a rectangle.
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2L + 2B \]
where \( L \) is the length and \( B \) is the breadth.

### Step 2: Substitute the known values into the perimeter formula.
We know that the perimeter \( P = 82 \) cm and the breadth \( B = 20 \) cm. Substituting these values into the formula gives:
\[ 82 = 2L + 2(20) \]

### Step 3: Simplify the equation.
Now, simplify the equation:
\[ 82 = 2L + 40 \]

### Step 4: Isolate the length \( L \).
To find \( L \), subtract 40 from both sides:
\[ 82 - 40 = 2L \]
\[ 42 = 2L \]

### Step 5: Solve for \( L \).
Now, divide both sides by 2:
\[ L = \frac{42}{2} = 21 \text{ cm} \]

### Step 6: Use the Pythagorean theorem to find the diagonal \( D \).
The diagonal \( D \) of a rectangle can be calculated using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]
Substituting the values of \( L \) and \( B \):
\[ D = \sqrt{21^2 + 20^2} \]

### Step 7: Calculate \( D \).
First, calculate \( 21^2 \) and \( 20^2 \):
\[ 21^2 = 441 \]
\[ 20^2 = 400 \]
Now, add these values:
\[ D = \sqrt{441 + 400} = \sqrt{841} \]

### Step 8: Find the square root.
Now, find the square root of 841:
\[ D = 29 \text{ cm} \]

### Final Answer:
The length of the diagonal of the rectangle is \( 29 \) cm.

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