If diagonals of a rhombus are 36 cm and 48 cm, then what is the perimeter (in cm) of the rhombus ?

Options

A. 30
B. 60
C. 120
D. 240

Answer

To find the perimeter 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 \(d_1 = 36 \, \text{cm}\) and \(d_2 = 48 \, \text{cm}\).

2. **Calculate the half lengths of the diagonals**:
- The diagonals of a rhombus bisect each other at right angles. Therefore, we can find the lengths of half of each diagonal:
- \(AO = \frac{d_1}{2} = \frac{36}{2} = 18 \, \text{cm}\)
- \(BO = \frac{d_2}{2} = \frac{48}{2} = 24 \, \text{cm}\)

3. **Use the Pythagorean theorem to find the length of a side of the rhombus**:
- In triangle \(AOB\), where \(O\) is the intersection point of the diagonals, we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
Substituting the values we found:
\[ AB^2 = 18^2 + 24^2 \]
\[ AB^2 = 324 + 576 = 900 \]
\[ AB = \sqrt{900} = 30 \, \text{cm} \]

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:
\[ P = 4 \times 30 = 120 \, \text{cm} \]

### Final Answer:
The perimeter of the rhombus is \(120 \, \text{cm}\).

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