Question

The diagonals of a rhombus are in the ratio 2:3. Find the length of smaller diagonal, if the area of the rhombus is 27 $c{m}^2.$
(4 marks)

Answer 1

Given:
Ratio of the diagonals = 2:3
Area of the rhombus = 27 cm${}^2$

Let the length of the smaller diagonal be $2y$ and the larger diagonal be $3y$
(0.5 marks)

Area of the rhombus
$=\frac{1}{2}\times$(product of diagonals)
(0.5 marks)

$\Rightarrow 27=\frac{1}{2}\times 2y\times 3y$

$\Rightarrow 27=\frac{1}{2}\times 6y^2$

$\Rightarrow 27=3y^2$

$\Rightarrow y^2=\frac{27}{3}=9$

$y$ = $\sqrt{9}$ = 3
(2 marks)

Hence, the length of the smaller diagonal = $2y$ = $2\left(3\right)$ = $6$ $cm$
(1 mark)