The perimeter of a rhombus is 52 cm. If one diagonal is 24 cm, find :
the length of its other diagonal,

Answer

To find the length of the other diagonal of a rhombus given its perimeter and one diagonal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the properties of a rhombus**:
A rhombus has all sides equal, and its diagonals bisect each other at right angles.

2. **Calculate the length of one side of the rhombus**:
The perimeter of the rhombus is given as 52 cm. Since the perimeter \( P \) of a rhombus is calculated as:
\[ P = 4 \times \text{side length} (S) \]
We can find the side length \( S \) by rearranging the formula:
\[ S = \frac{P}{4} = \frac{52}{4} = 13 \text{ cm} \]

3. **Identify the diagonals**:
Let the diagonals be \( AC \) and \( BD \). We know that \( BD = 24 \) cm.

4. **Use the property of diagonals**:
The diagonals bisect each other at point \( O \). Therefore, we have:
\[ AO = \frac{AC}{2} \quad \text{and} \quad BO = \frac{BD}{2} = \frac{24}{2} = 12 \text{ cm} \]

5. **Apply the Pythagorean theorem**:
In triangle \( AOB \), we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
We know \( AB = 13 \) cm (the side of the rhombus), so:
\[ 13^2 = AO^2 + 12^2 \]
Simplifying this gives:
\[ 169 = AO^2 + 144 \]

6. **Solve for \( AO \)**:
Rearranging the equation:
\[ AO^2 = 169 - 144 = 25 \]
Taking the square root:
\[ AO = \sqrt{25} = 5 \text{ cm} \]

7. **Find the length of diagonal \( AC \)**:
Since \( AO = \frac{AC}{2} \), we can find \( AC \):
\[ AC = 2 \times AO = 2 \times 5 = 10 \text{ cm} \]

### Final Answer:
The length of the other diagonal \( AC \) is **10 cm**.

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