Find the area of a rhombus having each side equal to 20 cm and one of its diagonals equal to 24 cm .

Answer

To find the area of a rhombus with each side measuring 20 cm and one diagonal measuring 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: Draw the rhombus and label it
Let's label the rhombus as ABCD, where AB = BC = CD = DA = 20 cm and diagonal AC = 24 cm.

### Step 3: Find half of diagonal AC
Since diagonal AC is 24 cm, half of diagonal AC (AO) will be:
\[ AO = \frac{AC}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find half of diagonal BD
In triangle AOB, we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
Where:
- \( AB = 20 \text{ cm} \)
- \( AO = 12 \text{ cm} \)
- \( BO \) is what we need to find.

Substituting the known values:
\[ 20^2 = 12^2 + BO^2 \]
\[ 400 = 144 + BO^2 \]

### Step 5: Solve for BO
Rearranging the equation gives:
\[ BO^2 = 400 - 144 = 256 \]
Taking the square root:
\[ BO = \sqrt{256} = 16 \text{ cm} \]

### Step 6: Find the length of diagonal BD
Since BO is half of diagonal BD, we can find BD:
\[ BD = 2 \times BO = 2 \times 16 = 32 \text{ cm} \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of the rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times AC \times BD \]
Substituting the values we found:
\[ A = \frac{1}{2} \times 24 \times 32 \]
\[ A = 12 \times 32 = 384 \text{ cm}^2 \]

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

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