A square has each side of 5cm. Find the length of one of its diagonals.

Answer

To find the length of one of the diagonals of a square with each side measuring 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the properties of the square**:
- A square has four equal sides and each angle is 90 degrees.

2. **Label the square**:
- Let's label the square as ABCD, where AB, BC, CD, and DA are the sides of the square.

3. **Use the side length**:
- Given that each side of the square is 5 cm, we have:
- AB = 5 cm
- BC = 5 cm
- CD = 5 cm
- DA = 5 cm

4. **Consider the diagonal**:
- We need to find the length of diagonal AC (or BD, since both diagonals are equal).

5. **Apply the Pythagorean theorem**:
- In triangle ADC (which is a right-angled triangle), we can apply the Pythagorean theorem:
\[ AC^2 = AD^2 + DC^2 \]
- Here, AD = 5 cm and DC = 5 cm.

6. **Substitute the values**:
- Substitute the lengths into the equation:
\[ AC^2 = 5^2 + 5^2 \]
- This simplifies to:
\[ AC^2 = 25 + 25 \]
\[ AC^2 = 50 \]

7. **Find the length of the diagonal**:
- To find AC, take the square root of both sides:
\[ AC = \sqrt{50} \]
- We can simplify \(\sqrt{50}\):
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \]

8. **Conclusion**:
- Therefore, the length of one of the diagonals of the square is:
\[ AC = 5\sqrt{2} \text{ cm} \]