If the diagonals of a rhombus are 24 cm and 10 cm. Then the perimeter of the rhombus is :

Question

Accepted Answer

To determine the perimeter of a rhombus when the lengths of its diagonals are given, we can use the properties of a rhombus and the Pythagorean theorem.

### Explanation:

1. **Properties of a Rhombus:**
   - A rhombus is a type of quadrilateral where all four sides are of equal length.
   - The diagonals of a rhombus bisect each other at right angles.

2. **Using the Diagonals:**
   - Given the diagonals are 24 cm and 10 cm, they bisect each other, forming four right-angled triangles within the rhombus.
   - Each half of the diagonals will be:
     - Half of 24 cm = 12 cm
     - Half of 10 cm = 5 cm

3. **Applying the Pythagorean Theorem:**
   - In one of the right-angled triangles formed by the diagonals, the two halves of the diagonals are the legs, and the side of the rhombus is the hypotenuse.
   - Using the Pythagorean theorem: 
     $$
     \text{Side}^2 = 12^2 + 5^2
     $$
     $$
     \text{Side}^2 = 144 + 25
     $$
     $$
     \text{Side}^2 = 169
     $$
     $$
     \text{Side} = \sqrt{169} = 13 \text{ cm}
     $$

4. **Calculating the Perimeter:**
   - Since all sides of a rhombus are equal, the perimeter $ P $ is given by:
     $$
     P = 4 \times \text{Side} = 4 \times 13 = 52 \text{ cm}
     $$

### Justification for Option 2:

- **Option 2 (52 cm)** is correct because, as calculated, the perimeter of the rhombus is indeed 52 cm.

### Explanation for Incorrect Options:

- **Option 1 (48 cm):** This option is incorrect because it suggests a side length of 12 cm, which does not satisfy the Pythagorean theorem with the given diagonals.
  
- **Option 3 (56 cm):** This option is incorrect because it implies a side length of 14 cm, which is not possible with the given diagonal lengths.

- **Option 4 (60 cm):** This option is incorrect because it suggests a side length of 15 cm, which also does not fit the conditions set by the diagonals.

In conclusion, the correct perimeter of the rhombus, based on the given diagonal lengths, is 52 cm, which aligns with Option 2.