The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is k cm. The value of 3k is

Answer

To solve the problem, we need to find the value of \(3k\) where \(k\) is the length of each side of the rhombus. Given the lengths of the diagonals of the rhombus are 16 cm and 12 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diagonals**: Let the diagonals \(AC\) and \(BD\) be given as:
- \(AC = 16 \, \text{cm}\)
- \(BD = 12 \, \text{cm}\)

2. **Find half of each diagonal**: Since the diagonals of a rhombus bisect each other at right angles, we can find the lengths of the segments formed by the intersection point \(O\):
- \(OA = OC = \frac{AC}{2} = \frac{16}{2} = 8 \, \text{cm}\)
- \(OB = OD = \frac{BD}{2} = \frac{12}{2} = 6 \, \text{cm}\)

3. **Use the Pythagorean theorem**: In triangle \(OAB\), we can apply the Pythagorean theorem to find the length of side \(AB\):
\[ AB^2 = OA^2 + OB^2 \]
Substituting the values:
\[ AB^2 = 8^2 + 6^2 = 64 + 36 = 100 \]

4. **Calculate the length of side \(AB\)**:
\[ AB = \sqrt{100} = 10 \, \text{cm} \]

5. **Identify \(k\)**: Since all sides of a rhombus are equal, we have:
\[ k = AB = 10 \, \text{cm} \]

6. **Calculate \(3k\)**:
\[ 3k = 3 \times 10 = 30 \, \text{cm} \]

### Final Answer:
The value of \(3k\) is \(30 \, \text{cm}\).

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