Question

The area of a rhombus is $24c{m}^2$ and the length of one diagonal is $2cm$ more than the other. Construct the rhombus and measure the length of one side.

• A. 6 cm
• B. 5 cm
• C. 4 cm
• D. 3 cm

Answer 1

The correct option is B 5 cm

Let's draw a rough sketch. ABCD be the rhombus where AC and BD are the diagonals.

• Area of the rhombus is given to be $24c{m}^2$.
• Let the length of one diagonal to be $xcm$. The length of other diagonal will be $(x+2)cm$.

$\because$ Area of rhombus is $\frac{1}{2}\times \text{product of diagonals}$
$\Rightarrow 24=\frac{\left(x\right)(x+2)}{2}$
$\Rightarrow \left(x\right)(x+2)=24\times 2=48$
$\Rightarrow x^2+2x=48$
$\Rightarrow x^2+2x-48=0$
$\Rightarrow x^2+8x-6x-48=0$
$\Rightarrow x(x+8)-6(x+8)=0$
$\Rightarrow (x-6)(x+8)=0$
Now,
$x=6$ or $x=-8$.
Negative values cannot be the length of the diagonals. Hence, $x=6$.
If the measure of $BD=6cm$, then measure of $AC=6+2=8cm.$

Steps of construction:

1. Draw the diagonal AC of length $8cm$.
2. Construct a perpendicular bisector for AC. Mark the intersecting point to be O.
3. Length of $BD=6cm$. Then, $BO=OD=3cm.$ Measure $3cm$ on the perpendicular lines from the point O.
4. Mark that points to be B and D.
5. Join AB, BC, CD and DA and, measure their lengths.


The measure of the all sides of the rhombus ABCD will be $5cm$.