Question

What is the area of a rhombus (in sq.m.) whose perimeter is 40 m and one of its diagonals is 16 m?

• A. \(88\)
• B. \(92\)
• C. \(96\)
• D. \(108\)

Hint:

For a rhombus, \[ \text{Area}=\frac12 d_1d_2. \] If one diagonal and side are known, use Pythagoras theorem to find the second diagonal.

Answer 1

The Correct Option is C

Step 1: Find the side of the rhombus. Perimeter: \[ 40 \] Since all sides are equal, \[ \text{Side} = \frac{40}{4} = 10 \] m.

Step 2: Use diagonal properties of a rhombus. Given one diagonal: \[ d_1=16 \] Half diagonal: \[ \frac{16}{2}=8 \] Let half of the other diagonal be \(x\). The diagonals bisect each other at right angles. Thus, \[ 10^2=8^2+x^2 \] \[ 100=64+x^2 \] \[ x^2=36 \] \[ x=6 \] Hence, \[ d_2=12 \]

Step 3: Find area. \[ \text{Area} = \frac12 d_1d_2 \] \[ = \frac12(16)(12) \] \[ =96 \] Therefore, \[ \boxed{96} \]