Find the area of a rhombus each side of which measures 20 cm and one of whose diagonal is 24 cm 

Answer

To find the area of a rhombus where each side measures 20 cm and one of the diagonals measures 24 cm, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has two diagonals that bisect each other at right angles. This means that each diagonal divides the rhombus into four right-angled triangles.

### Step 2: Identify the given values
- Side of the rhombus (AD, AB, CD) = 20 cm
- One diagonal (AC) = 24 cm

### Step 3: Calculate half of the diagonal AC
Since the diagonals bisect each other, we can find the lengths of the segments created by diagonal AC:
- OA = OC = AC / 2 = 24 cm / 2 = 12 cm

### Step 4: Use the Pythagorean theorem to find the other diagonal
In triangle AOD, we can apply the Pythagorean theorem:
- AD² = OA² + OD²
- Here, AD = 20 cm and OA = 12 cm.

Substituting the values:
- 20² = 12² + OD²
- 400 = 144 + OD²

### Step 5: Solve for OD²
Rearranging the equation:
- OD² = 400 - 144
- OD² = 256

### Step 6: Find the length of OD
Taking the square root:
- OD = √256 = 16 cm

### Step 7: Calculate the length of the other diagonal (BD)
Since OD = OB (because diagonals bisect each other):
- BD = OD + OB = OD + OD = 2 * OD = 2 * 16 cm = 32 cm

### Step 8: Use the area formula for the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

Substituting the values:
- \( A = \frac{1}{2} \times 24 \times 32 \)

### Step 9: Calculate the area
Calculating:
- \( A = \frac{1}{2} \times 24 \times 32 = 12 \times 32 = 384 \, \text{cm}^2 \)

### Final Answer
The area of the rhombus is \( 384 \, \text{cm}^2 \).

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