Question

The diagonals of a rhombus are 48 cm and 20 cm long. Find the perimeter of the rhombus.

Answer 1

Diagonals of a rhombus perpendicularly bisect each other. The statement can help us find a side of the rhombus. Consider the following figure.


ABCD is the rhombus and AC and BD are the diagonals. The diagonals intersect at point O.

We know:
$\angle DOC=90°$
$DO=OB=\frac{1}{2}DB=\frac{1}{2}\times 48=24\mathrm{cm}$
Similarly,
$AO=OC=\frac{1}{2}AC=\frac{1}{2}\times 20=10\mathrm{cm}$

Using Pythagoras' theorem in the right-angled triangle $∆$DOC, we get:

$$\begin{gathered} D{C}^2=\sqrt{D{O}^2+O{C}^2} \\ =\sqrt{{24}^2+{10}^2} \\ =\sqrt{576+100} \\ =\sqrt{676} \\ =26\mathrm{cm} \end{gathered}$$

DC is a side of the rhombus.
We know that in a rhombus, all sides are equal.
∴ Perimeter of ABCD$=26\times 4=104\mathrm{cm}$