The lengths of the diagonals of a rhombus are 24 cm and 32 cm. Find the length of the side of the rhombus.

Answer

To find the length of the side of a rhombus given the lengths of its diagonals, we can follow these steps:

### Step 1: Identify the lengths of the diagonals
Given:
- Diagonal AC = 24 cm
- Diagonal BD = 32 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 AO and BO:
- AO = AC / 2 = 24 cm / 2 = 12 cm
- BO = BD / 2 = 32 cm / 2 = 16 cm

### Step 3: Use the Pythagorean theorem
In triangle AOB, which is a right triangle:
- AO = 12 cm (one leg)
- BO = 16 cm (the other leg)
- AB = x cm (the hypotenuse, which is the side of the rhombus)

According to the Pythagorean theorem:
\[ AO^2 + BO^2 = AB^2 \]

### Step 4: Substitute the values into the equation
\[ 12^2 + 16^2 = x^2 \]
\[ 144 + 256 = x^2 \]

### Step 5: Calculate the sum
\[ 400 = x^2 \]

### Step 6: Find the value of x
To find x, take the square root of both sides:
\[ x = \sqrt{400} \]
\[ x = 20 \text{ cm} \]

### Conclusion
The length of each side of the rhombus is 20 cm.

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