If the diagonals of a rhombus are 8 and 6, then the square of its side is

Options

A. 25
B. 55
C. 64
D. 36

Answer

To find the square of the side of a rhombus when the lengths of its diagonals are given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths of the diagonals**:
- Let the lengths of the diagonals be \( d_1 = 8 \) units and \( d_2 = 6 \) units.

2. **Calculate 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 diagonals.
- Half of diagonal \( d_1 \):
\[ \frac{d_1}{2} = \frac{8}{2} = 4 \text{ units} \]
- Half of diagonal \( d_2 \):
\[ \frac{d_2}{2} = \frac{6}{2} = 3 \text{ units} \]

3. **Use the Pythagorean theorem**:
- In triangle \( OAB \) (where \( O \) is the intersection point of the diagonals), we can use the Pythagorean theorem to find the length of a side \( AB \) of the rhombus.
- According to the theorem:
\[ AB^2 = OA^2 + OB^2 \]
- Substituting the lengths we found:
\[ AB^2 = 4^2 + 3^2 \]
\[ AB^2 = 16 + 9 = 25 \]

4. **Find the length of the side**:
- Taking the square root to find the length of side \( AB \):
\[ AB = \sqrt{25} = 5 \text{ units} \]

5. **Calculate the square of the side**:
- Finally, the square of the side of the rhombus is:
\[ \text{Square of the side} = AB^2 = 25 \]

### Final Answer:
The square of the side of the rhombus is \( 25 \).

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