If the side of a rhombus is 20 meters and its shorter diagonal is three - fourths of its longer diagonal, then the area of the rhombus must be

Options

A. 375 `m^(2)`
B. 380 `m^(2)`
C. 384 `m^(2)`
D. 395 `m^(2)`

Answer

To find the area of the rhombus given the side length and the relationship between the diagonals, we can follow these steps:

### Step 1: Understand the relationship between the diagonals
Let the longer diagonal be \( d_1 \) and the shorter diagonal be \( d_2 \). According to the problem, we know that:
\[ d_2 = \frac{3}{4} d_1 \]

### Step 2: Use the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. Therefore, we can create a right triangle using half of each diagonal and the side of the rhombus. The half-lengths of the diagonals are:
\[ \frac{d_1}{2} \quad \text{and} \quad \frac{d_2}{2} \]

### Step 3: Substitute the relationship into the triangle
Substituting \( d_2 \) into the equation gives us:
\[ \frac{d_2}{2} = \frac{3}{4} \cdot \frac{d_1}{2} = \frac{3}{8} d_1 \]

### Step 4: Apply the Pythagorean theorem
Using the Pythagorean theorem in the right triangle formed by the sides and half-diagonals:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 20^2 \]
Substituting the values:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{3}{8} d_1\right)^2 = 20^2 \]

### Step 5: Solve for \( d_1 \)
This becomes:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{3}{8} d_1\right)^2 = 400 \]
Calculating each term:
\[ \frac{d_1^2}{4} + \frac{9}{64} d_1^2 = 400 \]
Finding a common denominator (64):
\[ \frac{16 d_1^2}{64} + \frac{9 d_1^2}{64} = 400 \]
This simplifies to:
\[ \frac{25 d_1^2}{64} = 400 \]
Multiplying both sides by 64:
\[ 25 d_1^2 = 25600 \]
Dividing by 25:
\[ d_1^2 = 1024 \]
Taking the square root:
\[ d_1 = 32 \text{ meters} \]

### Step 6: Calculate \( d_2 \)
Using the relationship \( d_2 = \frac{3}{4} d_1 \):
\[ d_2 = \frac{3}{4} \times 32 = 24 \text{ meters} \]

### Step 7: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values:
\[ A = \frac{1}{2} \times 32 \times 24 \]
Calculating:
\[ A = \frac{1}{2} \times 768 = 384 \text{ square meters} \]

### Final Answer
The area of the rhombus is **384 square meters**.