If the diagonal of a rhombus are 24 cm and 10 cm, then the area and perimeter of the rhombus are respectively

Options

A. 120 sq cm, 52 cm
B. 240 sq cm, 52 cm
C. 120 sq cm, 64 cm
D. 240 sq cm, 64 cm

Answer

To find the area and perimeter of a rhombus given its diagonals, we can follow these steps:

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

### Step 2: Calculate the lengths of half the diagonals
Since the diagonals of a rhombus bisect each other at right angles, we can find the lengths of half the diagonals:
- Half of diagonal 1 (d1/2) = 24 cm / 2 = 12 cm
- Half of diagonal 2 (d2/2) = 10 cm / 2 = 5 cm

### Step 3: Use the Pythagorean theorem to find the side length of the rhombus
We can consider one of the right triangles formed by the diagonals. The lengths of the half diagonals are the two legs of the triangle:
- One leg (a) = 12 cm (half of diagonal 1)
- Other leg (b) = 5 cm (half of diagonal 2)

Using the Pythagorean theorem:
\[ \text{Side length} (s) = \sqrt{(d1/2)^2 + (d2/2)^2} \]
\[ s = \sqrt{12^2 + 5^2} \]
\[ s = \sqrt{144 + 25} \]
\[ s = \sqrt{169} \]
\[ s = 13 \text{ cm} \]

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

### Step 5: Calculate the area of the rhombus
The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
\[ A = \frac{1}{2} \times 24 \text{ cm} \times 10 \text{ cm} \]
\[ A = \frac{1}{2} \times 240 \text{ cm}^2 \]
\[ A = 120 \text{ cm}^2 \]

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

---