A circle is passing through three vertices of a rhombus of side 8 cm and its centre is the fourth vertex of the thombus .Find the length of the longest diagonal of the rhombus (in cm) .

Options

A. `8sqrt(3)`
B. `4sqrt(3)`
C. `6sqrt(3)`
D. `2sqrt(3)`

Answer

To find the length of the longest diagonal of a rhombus with a side length of 8 cm, where a circle passes through three vertices and the center of the circle is at the fourth vertex, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Properties of the Rhombus**:
- A rhombus has all sides equal, and its diagonals bisect each other at right angles.
- Let the vertices of the rhombus be A, B, C, and D, where D is the center of the circle.

2. **Label the Sides**:
- Since each side of the rhombus is 8 cm, we have:
- AB = BC = CD = DA = 8 cm.

3. **Determine the Diagonal OB**:
- The diagonal OB is equal to the radius of the circle, which is the distance from the center (D) to vertex B.
- Therefore, OB = 8 cm.

4. **Use the Properties of the Diagonals**:
- Let the diagonals AC and BD intersect at point E.
- Since the diagonals bisect each other at right angles, we can denote:
- AE = EC = x (half of diagonal AC)
- BE = ED = 4 cm (half of diagonal BD, since OB = 8 cm).

5. **Apply the Pythagorean Theorem**:
- In triangle OBE, we can apply the Pythagorean theorem:
\[ OB^2 = OE^2 + BE^2 \]
- Substituting the known values:
\[ 8^2 = OE^2 + 4^2 \]
\[ 64 = OE^2 + 16 \]
\[ OE^2 = 64 - 16 = 48 \]
\[ OE = \sqrt{48} = 4\sqrt{3} \text{ cm} \]

6. **Calculate the Full Length of Diagonal AC**:
- Since AE = EC = OE, the full length of diagonal AC is:
\[ AC = AE + EC = 2 \times OE = 2 \times 4\sqrt{3} = 8\sqrt{3} \text{ cm} \]

7. **Determine the Length of the Longest Diagonal**:
- We have two diagonals: AC and BD.
- Length of diagonal BD = 8 cm.
- Length of diagonal AC = 8√3 cm.
- Since √3 is approximately 1.732, we can calculate:
\[ 8\sqrt{3} \approx 8 \times 1.732 \approx 13.856 \text{ cm} \]
- Therefore, the longest diagonal is AC.

### Final Answer:
The length of the longest diagonal of the rhombus is **8√3 cm**.