Question

If perimeter of a rhombus is 104 cm and length of one of its diagonals is 48 cm, then area of the rhombus (in cm²) is:

• A. 960
• B. 240
• C. 480
• D. 1344

Answer 1

The Correct Option is C

To find the area of a rhombus with a given perimeter and one diagonal, follow these steps:

1. Calculate the side length of the rhombus. Since the perimeter of a rhombus is four times the side length, the side length \( s \) is:
   \[ s = \frac{\text{Perimeter}}{4} = \frac{104}{4} = 26 \, \text{cm} \]
2. Use the properties of the rhombus to find the length of the other diagonal. Assume the diagonals of a rhombus are \( d_1 \) and \( d_2 \). We know:
   • \( d_1 = 48 \, \text{cm} \)
   • The diagonals bisect each other at right angles, dividing the rhombus into four right-angled triangles.
3. Using the Pythagorean theorem in one of the right triangles:
   \[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2 \]
   Substitute \( d_1 = 48 \, \text{cm} \) and \( s = 26 \, \text{cm} \) into the equation:
   \[ \left(\frac{48}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 26^2 \]
   \[ 24^2 + \left(\frac{d_2}{2}\right)^2 = 26^2 \]
   \[ 576 + \left(\frac{d_2}{2}\right)^2 = 676 \]
   Subtract 576 from both sides:
   \[ \left(\frac{d_2}{2}\right)^2 = 100 \]
   So,
   \[ \frac{d_2}{2} = 10 \]
   Therefore,
   \[ d_2 = 20 \, \text{cm} \]
4. Calculate the area of the rhombus using the formula:
   \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
   Substitute the values for the diagonals:
   \[ \text{Area} = \frac{1}{2} \times 48 \times 20 \]
   \[ \text{Area} = \frac{1}{2} \times 960 = 480 \, \text{cm}^2 \]

Hence, the area of the rhombus is 480 cm².