The diagonals of a rhombus are 18 cm and 24 cm. Find ,
(i) its area (ii) length of its sides (iii) its perimeter

Answer

To solve the problem step by step, we will find the area, length of the sides, and perimeter of the rhombus given its diagonals.

### Step 1: Find the Area of the Rhombus
The formula for the area of a rhombus when the lengths of the diagonals are known is:

\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]

where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

Given:
- \( d_1 = 24 \, \text{cm} \)
- \( d_2 = 18 \, \text{cm} \)

Substituting the values into the formula:

\[ \text{Area} = \frac{1}{2} \times 24 \times 18 \]

Calculating the area:

\[ \text{Area} = \frac{1}{2} \times 432 = 216 \, \text{cm}^2 \]

### Step 2: Find the Length of the Sides of the Rhombus
The diagonals of a rhombus bisect each other at right angles. Therefore, we can use the right triangle formed by half of each diagonal to find the length of a side.

Let \( O \) be the intersection point of the diagonals. Then:

- \( OA = \frac{d_1}{2} = \frac{24}{2} = 12 \, \text{cm} \)
- \( OB = \frac{d_2}{2} = \frac{18}{2} = 9 \, \text{cm} \)

Using the Pythagorean theorem in triangle \( AOB \):

\[ AB^2 = OA^2 + OB^2 \]

Substituting the values:

\[ AB^2 = 12^2 + 9^2 \]

Calculating:

\[ AB^2 = 144 + 81 = 225 \]

Taking the square root to find \( AB \):

\[ AB = \sqrt{225} = 15 \, \text{cm} \]

### Step 3: Find the Perimeter of the Rhombus
The perimeter \( P \) of a rhombus is given by:

\[ P = 4 \times \text{side} \]

Since all sides are equal and we found \( AB = 15 \, \text{cm} \):

\[ P = 4 \times 15 = 60 \, \text{cm} \]

### Summary of Results
(i) Area of the rhombus: \( 216 \, \text{cm}^2 \)
(ii) Length of each side: \( 15 \, \text{cm} \)
(iii) Perimeter of the rhombus: \( 60 \, \text{cm} \)

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