If the diagonals of a rhombus are 30 cm and 40 cm, then the length of side of rhombus is

Options

A. 20 cm
B. 22 cm
C. 25 cm
D. 45 cm

Answer

To find the length of the side of a rhombus when the lengths of its diagonals are given, we can use the properties of the rhombus and the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Understand the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. This means that each diagonal is divided into two equal parts, and they intersect at a right angle (90 degrees).

### Step 2: Identify the lengths of the diagonals
Given:
- Diagonal AC = 30 cm
- Diagonal BD = 40 cm

### Step 3: Find the lengths of the half-diagonals
Since the diagonals bisect each other:
- AO = OC = AC/2 = 30 cm / 2 = 15 cm
- BO = OD = BD/2 = 40 cm / 2 = 20 cm

### Step 4: Use the Pythagorean theorem
In triangle AOB, we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]

### Step 5: Substitute the values
Now, substitute the lengths of AO and BO into the equation:
\[ AB^2 = (15 \, \text{cm})^2 + (20 \, \text{cm})^2 \]

### Step 6: Calculate the squares
Calculate the squares:
\[ AB^2 = 225 \, \text{cm}^2 + 400 \, \text{cm}^2 \]

### Step 7: Add the results
Now, add the two results:
\[ AB^2 = 625 \, \text{cm}^2 \]

### Step 8: Take the square root
To find AB, take the square root of both sides:
\[ AB = \sqrt{625 \, \text{cm}^2} = 25 \, \text{cm} \]

### Conclusion
The length of each side of the rhombus is 25 cm.

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