The diagonals of a rhombus are 24 cm and 10 cm. Calculate its area and perimeter.

Answer

To solve the problem, we need to calculate the area and perimeter of a rhombus given its diagonals. Here are the steps to find the solution:

### Step 1: Identify the diagonals
The diagonals of the rhombus are given as:
- Diagonal 1 (d1) = 24 cm
- Diagonal 2 (d2) = 10 cm

### Step 2: Calculate the area of the rhombus
The formula for the area (A) of a rhombus using its diagonals is:
\[ A = \frac{d1 \times d2}{2} \]
Substituting the values:
\[ A = \frac{24 \times 10}{2} = \frac{240}{2} = 120 \text{ cm}^2 \]

### Step 3: Find the lengths of the half diagonals
Since the diagonals bisect each other at right angles, we can find the lengths of the half diagonals:
- Half of diagonal 1 (OD) = \( \frac{24}{2} = 12 \) cm
- Half of diagonal 2 (OC) = \( \frac{10}{2} = 5 \) cm

### Step 4: Use the Pythagorean theorem to find the side of the rhombus
In triangle DOC, we can apply the Pythagorean theorem:
\[ DC^2 = OD^2 + OC^2 \]
Substituting the values:
\[ DC^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Taking the square root:
\[ DC = \sqrt{169} = 13 \text{ cm} \]

### Step 5: Calculate the perimeter of the rhombus
The perimeter (P) of a rhombus is given by:
\[ P = 4 \times \text{side} \]
Substituting the side length:
\[ P = 4 \times 13 = 52 \text{ cm} \]

### Final Results
- Area of the rhombus = 120 cm²
- Perimeter of the rhombus = 52 cm

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