If the length of the diagonal of a square is 20cm,then its perimeter must be

Options

A. A. `40sqrt2 cm`
B. B. `30sqrt2 cm`
C. C. `10cm`
D. D. `15sqrt2 cm`

Answer

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

### Step 1: Understand the relationship between the diagonal and the sides of the square.
The diagonal \(d\) of a square is related to its side length \(a\) by the formula:
\[ d = a\sqrt{2} \]

### Step 2: Substitute the given diagonal length into the formula.
We are given that the diagonal \(d\) is 20 cm. We can substitute this value into the formula:
\[ 20 = a\sqrt{2} \]

### Step 3: Solve for the side length \(a\).
To find \(a\), we can rearrange the equation:
\[ a = \frac{20}{\sqrt{2}} \]

### Step 4: Simplify the expression for \(a\).
To simplify \(\frac{20}{\sqrt{2}}\), we can multiply the numerator and the denominator by \(\sqrt{2}\):
\[ a = \frac{20\sqrt{2}}{2} = 10\sqrt{2} \]

### Step 5: Calculate the perimeter of the square.
The perimeter \(P\) of a square is given by the formula:
\[ P = 4a \]
Substituting \(a = 10\sqrt{2}\) into the perimeter formula:
\[ P = 4 \times 10\sqrt{2} = 40\sqrt{2} \]

### Final Answer:
The perimeter of the square is \(40\sqrt{2}\) cm.

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