Question

A wire is in the form of a circle of radius 35 cm. If it is bent into the shape of a rhombus, what is the side of the rhombus?
A) $32cm$
B) $70cm$
C) $55cm$
D) $17cm$

Answer 1

Hint:
We can find the length of the wire by finding the circumference of the circle. As we bend the same wire to a rhombus, the perimeter of the rhombus will be also equal to the circumference of the circle. Then we can find the side of the rhombus by dividing the perimeter with 4 as all the sides of a rhombus are equal.

Complete step by step solution:

We are given that the radius of the circle is 35cm. Using the radius, we can find the circumference of the circle using the equation,
$C = 2\pi r$
On substituting the values, we get,
$ \Rightarrow C = 2 \times \dfrac{{22}}{7} \times 35$
On simplification we get,
$ \Rightarrow C = 2 \times 22 \times 5$
On multiplication we get,
$ \Rightarrow C = 220cm$
Thus, the circumference of the circle is 220cm.
Now we can consider the rhombus. We know that for a rhombus, all the four sides will be equal.
Let a be the length of the sides of the rhombus.
Then its perimeter is given by the sum of the sides.
\[ \Rightarrow P = a + a + a + a\]
So we have,
\[ \Rightarrow P = 4a\]
It is given that the wire in the form of the circle is bent into the shape of rhombus. So, their perimeter will be equal.
$ \Rightarrow C = P$
$ \Rightarrow 220cm = 4a$
On dividing throughout with 4, we get,
$ \Rightarrow a = \dfrac{{220}}{4}cm$
On simplification we get,
$ \Rightarrow a = 55cm$
Therefore, the length of the side is $55cm$ .

So, the correct answer is option C, $55cm$.

Note:
A rhombus is a quadrilateral with all the four sides equal. A rhombus cannot be considered as a regular quadrilateral as the angles are not equal. The difference between the rhombus and the square is that a square has all the four angles equal to $90^\circ $ and the rhombus can have any measure of angle. So, we can say that a square is a rhombus but the rhombus is not a square.
We must take the value of $\pi $ as $\dfrac{{22}}{7}$ because the radius is a multiple of 7 and the denominator can be cancelled. So, we can avoid decimals in the circumference. We must only equate the perimeter of the circle and rhombus, not the area.