Find the perimeter of the rectangle whose length is `40 cm` and a diagonal is `41 cm`.

Options

A. `97cm`
B. `89cm`
C. `98cm`
D. `88cm`

Explanation

To find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm, we can follow these steps:

### Step 1: Understand the relationship in a rectangle
In a rectangle, the diagonal divides it into two right-angled triangles. We can use the Pythagorean theorem to find the breadth (width) of the rectangle. According to the theorem, for a right triangle with sides \(a\) and \(b\), and hypotenuse \(c\), the relationship is given by:
\[ c^2 = a^2 + b^2 \]

### Step 2: Assign values
Here, let:
- Length (l) = 40 cm
- Diagonal (d) = 41 cm
- Breadth (b) = ?

### Step 3: Apply the Pythagorean theorem
Using the Pythagorean theorem:
\[ d^2 = l^2 + b^2 \]

Substituting the known values:
\[ 41^2 = 40^2 + b^2 \]

### Step 4: Calculate the squares
Calculate \(41^2\) and \(40^2\):
- \(41^2 = 1681\)
- \(40^2 = 1600\)

### Step 5: Substitute and solve for breadth
Now substitute these values into the equation:
\[ 1681 = 1600 + b^2 \]

Rearranging gives:
\[ b^2 = 1681 - 1600 \]
\[ b^2 = 81 \]

### Step 6: Find the breadth
Taking the square root of both sides:
\[ b = \sqrt{81} = 9 \text{ cm} \]

### Step 7: Calculate the perimeter
The formula for the perimeter \(P\) of a rectangle is:
\[ P = 2 \times (l + b) \]

Substituting the values of length and breadth:
\[ P = 2 \times (40 + 9) \]
\[ P = 2 \times 49 \]
\[ P = 98 \text{ cm} \]

### Final Answer
The perimeter of the rectangle is **98 cm**.

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