Question

The diagonals of rhombus measure 16cm and 30cm. Find its perimeter.

Answer 1

Hint: The diagonals of rhombus bisect each other and perpendicular on each other. By using this statement we can find out the rhombus side. And every side is equal in rhombus by its definition. Once we have all sides then we can figure out its perimeter.

Complete step-by-step answer:
It is given that the diagonal of rhombus are 16cm and 30cm. it will bisect each other and will be perpendicular on each other.
Draw a rhombus ABCD along with its diagonal which cut each other on O.

In above figure AC = 30cm and BD = 16cm.
AC and BD are bisecting each other, so AO = 15cm and OD = 8cm.
And \[\angle AOB={{90}^{\circ }}\]
Now, we have a triangle AOB which is a right angled triangle so we can use Pythagoras theorem in order to find side AD (which is a hypotenuse in triangle AOB).
Pythagoras theorem states,
In a right angled triangle, the square of hypotenuse is equal to the sum of square of its base and square of its height.
  $ \begin{align}
  & A{{D}^{2}}=O{{A}^{2}}+O{{D}^{2}} \\
 & A{{D}^{2}}={{15}^{2}}+{{8}^{2}} \\
 & A{{D}^{2}}=225+64 \\
 & A{{D}^{2}}=289 \\
 & AD=17 \\
\end{align} $
Here, we will ignore -17 because the side of the triangle cannot be negative.
All sides are equal in a rhombus. So, the perimeter of rhombus is,
  $ 4\times side $
 \[\begin{matrix}
  Perimeter=4\times side \\
  =4\times AD \\
  =4\times 17 \\
  =68 \\
\end{matrix}\]
The perimeter of rhombus is 68 units.

Note:- Here, we can also find the relation between rhombus’s diagonals and its area in order to find its perimeter. Though Square’s all sides are equal and it has 4 right angles but it is not necessary in rhombus. We can say that the square is a kind of rhombus.