The length of a side of a rhombus is 10 m and one of its diagonal is 12 m. The length of the other diagonal is :

Options

A. 15 m
B. 18 m
C. 16 m
D. can't be determined

Answer

To solve the problem, we need to find the length of the other diagonal of the rhombus given that one diagonal is 12 m and the length of each side is 10 m.

### Step-by-Step Solution:

1. **Identify the properties of the rhombus**:
A rhombus has diagonals that bisect each other at right angles. Let’s denote the diagonals as AC and BD, where AC = 12 m (given) and we need to find BD.

2. **Find half of the diagonal AC**:
Since the diagonals bisect each other, we can find AO (half of diagonal AC):
\[ AO = \frac{AC}{2} = \frac{12}{2} = 6 \text{ m} \]

3. **Use the Pythagorean theorem**:
In triangle AOB, we can apply the Pythagorean theorem. Here, AB is the side of the rhombus (10 m), AO is half of diagonal AC (6 m), and BO is half of diagonal BD (which we will denote as \( x \)):
\[ AB^2 = AO^2 + BO^2 \]
Substituting the known values:
\[ 10^2 = 6^2 + x^2 \]
\[ 100 = 36 + x^2 \]

4. **Solve for \( x^2 \)**:
Rearranging the equation gives:
\[ x^2 = 100 - 36 = 64 \]

5. **Find \( x \)**:
Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \text{ m} \]

6. **Calculate the full length of diagonal BD**:
Since \( x \) is half of diagonal BD (BO), the full length of diagonal BD is:
\[ BD = 2x = 2 \times 8 = 16 \text{ m} \]

### Final Answer:
The length of the other diagonal (BD) is **16 m**.

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