The length of each of the sides of a rhombus is given as 5 m and the length of one of its diagonals is 2.8 m Find the area of the rhombus.

Options

A. `7 m^(2)`
B. `13.44 m^(2)`
C. `6.72 m^(2)`
D. `14 m^(2)`

Answer

To find the area of the rhombus given the length of each side and one diagonal, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has four equal sides, and its diagonals bisect each other at right angles. This means that if we know one diagonal, we can find the other diagonal using the Pythagorean theorem.

### Step 2: Given values
- Length of each side of the rhombus (AB = BC = CD = DA) = 5 m
- Length of one diagonal (BD) = 2.8 m

### Step 3: Find half of the diagonal
Since the diagonals bisect each other, we can find half of diagonal BD:
\[ BO = OD = \frac{BD}{2} = \frac{2.8}{2} = 1.4 \, \text{m} \]

### Step 4: Use the Pythagorean theorem
In triangle BOC (where O is the intersection of the diagonals), we can apply the Pythagorean theorem:
- Let OC be the half of the other diagonal AC.
- We know:
\[ BC^2 = BO^2 + OC^2 \]
\[ 5^2 = (1.4)^2 + OC^2 \]

### Step 5: Calculate OC
Substituting the known values:
\[ 25 = 1.96 + OC^2 \]
\[ OC^2 = 25 - 1.96 = 23.04 \]
\[ OC = \sqrt{23.04} = 4.8 \, \text{m} \]

### Step 6: Find the full length of the second diagonal
Since OC is half of diagonal AC:
\[ AC = 2 \times OC = 2 \times 4.8 = 9.6 \, \text{m} \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times (BD) \times (AC) \]
Substituting the values we have:
\[ A = \frac{1}{2} \times 2.8 \times 9.6 \]
\[ A = \frac{1}{2} \times 26.88 = 13.44 \, \text{m}^2 \]

### Final Answer
The area of the rhombus is \( 13.44 \, \text{m}^2 \).

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