The perimeter of a rhombus is 146 cm and one of its diagonal is 55 cm. Find the other diagonal and the area of the rhombus.

Options

A. 24 cm, 660 `cm^2`
B. 24 cm, 330 `cm^(2)`
C. 48 cm , 660 `cm^(2)`
D. 48 cm, 1320 `cm^(2)`

Answer

To solve the problem, we need to find the other diagonal (D2) of the rhombus and then calculate its area.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- Perimeter of the rhombus (P) = 146 cm
- One diagonal (D1) = 55 cm

2. **Using the Perimeter Formula:**
The perimeter of a rhombus is given by the formula:
\[ P = 4a \]
where \( a \) is the length of one side of the rhombus. Therefore, we can find the side length \( a \):
\[ a = \frac{P}{4} = \frac{146}{4} = 36.5 \text{ cm} \]

3. **Using the Diagonal Relationship:**
The relationship between the diagonals and the side of the rhombus is given by:
\[ a^2 = \left(\frac{D1}{2}\right)^2 + \left(\frac{D2}{2}\right)^2 \]
Substituting the known values:
\[ (36.5)^2 = \left(\frac{55}{2}\right)^2 + \left(\frac{D2}{2}\right)^2 \]
This simplifies to:
\[ 1332.25 = 30.25^2 + \left(\frac{D2}{2}\right)^2 \]
\[ 1332.25 = 915.0625 + \left(\frac{D2}{2}\right)^2 \]

4. **Solving for D2:**
Rearranging the equation gives:
\[ \left(\frac{D2}{2}\right)^2 = 1332.25 - 915.0625 \]
\[ \left(\frac{D2}{2}\right)^2 = 417.1875 \]
Taking the square root of both sides:
\[ \frac{D2}{2} = \sqrt{417.1875} \approx 20.43 \]
Therefore:
\[ D2 \approx 2 \times 20.43 \approx 40.86 \text{ cm} \]

5. **Calculating the Area of the Rhombus:**
The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times D1 \times D2 \]
Substituting the values we have:
\[ A = \frac{1}{2} \times 55 \times 40.86 \]
\[ A = \frac{1}{2} \times 2247.3 \approx 1123.65 \text{ cm}^2 \]

### Final Answers:
- The other diagonal (D2) is approximately **40.86 cm**.
- The area of the rhombus is approximately **1123.65 cm²**.