ABCD is a rhombus with side 10 cm and one of its diagonals BD equal to 12 cm. The area of the square constructed on the other diagonal AC as its side is (in `cm^(2)`)

Options

A. 144
B. 196
C. 256
D. 400

Answer

To solve the problem, we need to find the area of the square constructed on the diagonal AC of the rhombus ABCD, given that the side of the rhombus is 10 cm and one of its diagonals, BD, is 12 cm.

### Step-by-Step Solution:

1. **Identify the properties of the rhombus**:
- In a rhombus, the diagonals bisect each other at right angles.
- Let the diagonals be AC and BD. We know that BD = 12 cm.

2. **Find the half-length of diagonal BD**:
- Since the diagonals bisect each other, half of diagonal BD is:
\[ \frac{BD}{2} = \frac{12}{2} = 6 \text{ cm} \]

3. **Use the Pythagorean theorem**:
- Let the half-length of diagonal AC be \( x \). The sides of the rhombus form right triangles with the halves of the diagonals.
- According to the Pythagorean theorem:
\[ \text{(side of rhombus)}^2 = \left(\frac{BD}{2}\right)^2 + \left(\frac{AC}{2}\right)^2 \]
- Plugging in the known values:
\[ 10^2 = 6^2 + x^2 \]
- This simplifies to:
\[ 100 = 36 + x^2 \]
- Rearranging gives:
\[ x^2 = 100 - 36 = 64 \]
- Therefore:
\[ x = \sqrt{64} = 8 \text{ cm} \]

4. **Calculate the full length of diagonal AC**:
- Since \( x \) is half of diagonal AC, the full length of diagonal AC is:
\[ AC = 2x = 2 \times 8 = 16 \text{ cm} \]

5. **Find the area of the square constructed on diagonal AC**:
- The area \( A \) of a square with side length \( s \) is given by:
\[ A = s^2 \]
- Here, the side of the square is the length of diagonal AC, which is 16 cm:
\[ A = 16^2 = 256 \text{ cm}^2 \]

### Final Answer:
The area of the square constructed on diagonal AC is **256 cm²**.