Question

The diagonals of a rhombus are in the ratio 3:4. Find the length of smaller diagonal, if the area of the rhombus is 54 $c{m}^2.$
(3 marks)

Answer 1

Given:
Ratio of the diagonals = 3:4
Area of the rhombus = 54 cm${}^2$

Let the length of the smaller diagonal be $3y$ and the larger diagonal be $4y$
(0.5 marks)

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

$\Rightarrow 54=\frac{1}{2}\times 3y\times 4y$

$\Rightarrow 54=\frac{1}{2}\times 12y^2$

$\Rightarrow 54=6y^2$

$\Rightarrow y^2=\frac{54}{6}=9$

$y$ = $\sqrt{9}$ = 3
(1 mark)

Hence, the length of the smaller diagonal = $3y$ = $3\left(3\right)$ = $9$ $cm$
(0.5 marks)