The diagonals of a rhombus are 10 cm and 24 cm. Find the length of a side of the rhombus.

Answer

To find the length of a side of the rhombus with diagonals measuring 10 cm and 24 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diagonals**: Let the lengths of the diagonals be \(d_1 = 10 \, \text{cm}\) and \(d_2 = 24 \, \text{cm}\).

2. **Calculate half of each diagonal**: Since the diagonals of a rhombus bisect each other at right angles, we find:
- Half of \(d_1\): \(\frac{d_1}{2} = \frac{10}{2} = 5 \, \text{cm}\)
- Half of \(d_2\): \(\frac{d_2}{2} = \frac{24}{2} = 12 \, \text{cm}\)

3. **Form a right triangle**: The halves of the diagonals form a right triangle with the sides being \(5 \, \text{cm}\) and \(12 \, \text{cm}\). The hypotenuse of this triangle will be the length of a side of the rhombus.

4. **Apply the Pythagorean theorem**: According to the Pythagorean theorem, we can find the hypotenuse \(h\) (which is the side of the rhombus) using the formula:
\[ h^2 = \text{(base)}^2 + \text{(height)}^2 \]
Substituting the values:
\[ h^2 = 5^2 + 12^2 \]
\[ h^2 = 25 + 144 \]
\[ h^2 = 169 \]

5. **Calculate the length of the side**: Taking the square root of both sides gives:
\[ h = \sqrt{169} = 13 \, \text{cm} \]

6. **Conclusion**: Therefore, the length of each side of the rhombus is \(13 \, \text{cm}\).