What will be the perimeter of square. If the sum of lengths of diagonal is 144 cm.

Options

A. 144 cm
B. `144V2 cm`
C. 288 cm
D. 228^" cm

Answer

To find the perimeter of a square when the sum of the lengths of its diagonals is given, we can follow these steps:

### Step 1: Understand the properties of the square
A square has four equal sides and two equal diagonals. The length of each diagonal can be calculated using the formula:
\[ D = a \sqrt{2} \]
where \( D \) is the length of the diagonal and \( a \) is the length of one side of the square.

### Step 2: Set up the equation for the diagonals
According to the problem, the sum of the lengths of the diagonals is 144 cm. Since both diagonals are equal, we can express this as:
\[ D + D = 144 \]
This simplifies to:
\[ 2D = 144 \]

### Step 3: Solve for the length of one diagonal
To find the length of one diagonal \( D \), we divide both sides of the equation by 2:
\[ D = \frac{144}{2} = 72 \text{ cm} \]

### Step 4: Relate the diagonal to the side of the square
Using the relationship between the diagonal and the side of the square, we can substitute \( D \) into the formula:
\[ D = a \sqrt{2} \]
Thus:
\[ 72 = a \sqrt{2} \]

### Step 5: Solve for the side length \( a \)
To find the side length \( a \), we rearrange the equation:
\[ a = \frac{72}{\sqrt{2}} \]

### Step 6: Rationalize the denominator
To simplify \( a \), we multiply the numerator and denominator by \( \sqrt{2} \):
\[ a = \frac{72 \sqrt{2}}{2} = 36 \sqrt{2} \text{ cm} \]

### Step 7: Calculate the perimeter of the square
The perimeter \( P \) of a square is given by:
\[ P = 4a \]
Substituting the value of \( a \):
\[ P = 4 \times 36 \sqrt{2} = 144 \sqrt{2} \text{ cm} \]

### Final Answer
The perimeter of the square is:
\[ \boxed{144 \sqrt{2} \text{ cm}} \]