Find the area of a rhombus in which each side is 15 cm long and one of whose diagonals is 24

Options

A. `211cm^(2)`
B. `216cm^(2)`
C. `218cm^(2)`
D. `217cm^(2)`

Answer

To find the area of a rhombus given the length of each side and one of its diagonals, 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 if we know one diagonal, we can find the other diagonal using the properties of right triangles formed by the diagonals.

### Step 2: Identify the given values
- Length of each side (s) = 15 cm
- Length of one diagonal (AC) = 24 cm

### Step 3: Find half of the given diagonal
Since the diagonals bisect each other, we can find half of diagonal AC:
- AO = AC / 2 = 24 cm / 2 = 12 cm

### Step 4: Use the Pythagorean theorem
In triangle AOD (where O is the intersection of the diagonals), we can apply the Pythagorean theorem to find the length of the other half diagonal (OD):
- We know:
- AO = 12 cm (half of diagonal AC)
- AD = 15 cm (the side of the rhombus)

Using the Pythagorean theorem:
\[ AD^2 = AO^2 + OD^2 \]
\[ 15^2 = 12^2 + OD^2 \]
\[ 225 = 144 + OD^2 \]

### Step 5: Solve for OD
Rearranging the equation gives:
\[ OD^2 = 225 - 144 \]
\[ OD^2 = 81 \]
Taking the square root:
\[ OD = 9 \text{ cm} \]

### Step 6: Find the length of the second diagonal
Since OD is half of the second diagonal (BD), we can find the full length of diagonal BD:
- BD = 2 * OD = 2 * 9 cm = 18 cm

### Step 7: Calculate the area of 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:
- \( d_1 = 24 \) cm (diagonal AC)
- \( d_2 = 18 \) cm (diagonal BD)

Substituting the values:
\[ A = \frac{1}{2} \times 24 \times 18 \]
\[ A = 12 \times 18 \]
\[ A = 216 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is **216 cm²**.

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