Question

A circular wire of radius 1 dm is cut and is placed along the circumference of a circle of radius one meter. The angle subtended by the wire at the center of the radius is equal to
 \[\begin{align}
  & A.\dfrac{\pi }{4}\text{ radian} \\
 & \text{B}\text{.}\dfrac{\pi }{3}\text{ radian} \\
 & \text{C}\text{.}\dfrac{\pi }{5}\text{ radian} \\
 & \text{D}\text{.}\dfrac{\pi }{10}\text{ radian} \\
\end{align}\]

Answer 1

Hint: To solve this question, we will assume that when the wire of radius 1 dm is cut and placed on the circumference of the circle of radius of one meter, its end points are A and B. When these points A and B are joined to the circumference of the circle of radius of 1m circle. After joining, it will form a sector, we have to determine the angle of that sector.

Complete step-by-step answer:
It is given that a circular wire of radius 1 dm is cut. The length of the wire after cutting the wire will be equal to the circumference of the circular wire. So, to determine the length of wire, we will have to find the circumference of the circle. The circumference of the circle with radius r is given by the formula shown below:
\[\text{Circumference}=\text{ 2}\pi \text{r}\]
On applying this formula, we get:
\[\text{Length of wire = Circumference}=\text{ 2 }\times \text{ }\pi \text{ }\times \text{ }\left( 1dm \right)\]
We know that\[\text{1 dm}=0.\text{1 metre}\]. Thus we get:
\[\begin{align}
  & \Rightarrow \text{ length of wire = 2}\pi \left( 0.1 \right)\text{ metre} \\
 & \Rightarrow \text{ length of wire = }\dfrac{2\pi }{10}\text{ metre} \\
 & \Rightarrow \text{ length of wire = }\dfrac{\pi }{5}\text{ metre} \\
\end{align}\]
Now, let this wire have endpoints A and B. It is given that this wire AB is placed on the circumference of the circle of radius of 1m as shown in the figure:

Now, we have to determine the angle $\theta $ of the segment. For this first we will determine the circumference of this circle. Thus we have:
\[\text{Circumference}=2\pi \left( 1m \right)\]
\[\text{Circumference }=2\pi m\]
We know that the circumference subtends an angle of $2\pi $ or the center. So $2\pi m$ will subtend an angle of $2\pi $ radian. So the angle subtended by \[1m\text{ = }\dfrac{2\pi }{2\pi }\text{ radian = 1 radian}\text{.}\]
Similarly, the angle subtended by wire of length \[\dfrac{\pi }{5}m\] is \[\Rightarrow \text{ 1 radian }\times \text{ }\dfrac{\pi }{5}\text{ = }\dfrac{\pi }{5}\text{ radian}\]

Note: We have assumed that the length of the wire remains the same after cutting i.e. it does not contract or expand. We have also assumed that the thickness of the wire is negligible.