If the diagonals of a rhombus are 18 cm and 24 cm respectively , then its perimeter is :

Options

A. 15 cm
B. 42 cm
C. 60 cm
D. 70 cm

Answer

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

### Step 1: Identify the lengths of the diagonals
Given:
- Diagonal AC = 18 cm
- Diagonal BD = 24 cm

### Step 2: Find the lengths of half the diagonals
Since the diagonals of a rhombus bisect each other at right angles, we can find the lengths of half the diagonals:
- Half of diagonal AC = \( \frac{18}{2} = 9 \) cm
- Half of diagonal BD = \( \frac{24}{2} = 12 \) cm

### Step 3: Use the Pythagorean theorem to find the length of a side of the rhombus
In triangle AOD (where O is the intersection point of the diagonals), we can apply the Pythagorean theorem:
- Let AD be the side of the rhombus (hypotenuse).
- We have:
- Perpendicular (half of diagonal BD) = 12 cm
- Base (half of diagonal AC) = 9 cm

Using the Pythagorean theorem:
\[ AD^2 = AO^2 + OD^2 \]
\[ AD^2 = 12^2 + 9^2 \]
\[ AD^2 = 144 + 81 \]
\[ AD^2 = 225 \]
\[ AD = \sqrt{225} = 15 \text{ cm} \]

### Step 4: Calculate the perimeter of the rhombus
The perimeter (P) of a rhombus is given by the formula:
\[ P = 4 \times \text{side length} \]
Substituting the side length we found:
\[ P = 4 \times 15 = 60 \text{ cm} \]

### Conclusion
The perimeter of the rhombus is **60 cm**.

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