Find the length of the side of a square if the length of its diagonal is 12 cm .

Answer

To find the length of the side of a square when the length of its diagonal is given, we can use the properties of a square and the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Understand the properties of the square
A square has four equal sides and the angles between the sides are all 90 degrees. If we denote the length of each side of the square as \( x \), then we can visualize the square as follows:

```
A ---- B
| |
| |
D ---- C
```

### Step 2: Use the Pythagorean theorem
In a square, the diagonal divides it into two right-angled triangles. According to the Pythagorean theorem, the square of the length of the diagonal \( d \) is equal to the sum of the squares of the lengths of the sides:

\[ d^2 = x^2 + x^2 \]

Since both sides are equal, we can simplify this to:

\[ d^2 = 2x^2 \]

### Step 3: Substitute the given diagonal length
We know the diagonal \( d \) is 12 cm. Therefore, we can substitute this value into the equation:

\[ 12^2 = 2x^2 \]

Calculating \( 12^2 \):

\[ 144 = 2x^2 \]

### Step 4: Solve for \( x^2 \)
Now, divide both sides by 2 to isolate \( x^2 \):

\[ x^2 = \frac{144}{2} \]

Calculating the right side gives:

\[ x^2 = 72 \]

### Step 5: Find \( x \)
To find \( x \), take the square root of both sides:

\[ x = \sqrt{72} \]

### Step 6: Simplify \( \sqrt{72} \)
We can simplify \( \sqrt{72} \) as follows:

\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]

### Conclusion
Thus, the length of each side of the square is:

\[ \boxed{6\sqrt{2} \text{ cm}} \]