Question

All the arcs in the following diagram are semi-circles. This diagram shows two paths connecting $A$ to $B$. Path $1$ is the single large semi-circle and Path $2$ consists of the chain of small semi-circles. Then

• A. Path $1$ is longer than Path $2$
• B. Path $1$ is of the same length as Path $2$
• C. Path $1$ is shorter than Path $2$
• D. Path $1$ is of the same length as Path $2$, only if the number of semi-circles is not more than $4$.

Answer 1

Answer: (B)

Path $1$ is of the same length as Path $2$

Let there are $n$ small semi-circles in the chain of small semicircles with radius $r$.
$\therefore$ Length of single large semi-circle $=$ $\frac{1}{2}\left(2\pi nr\right)$ $=$ $n\pi r$
Path $1$ $=$ $n\pi r$
Length of small circles $=$ $n\times \frac{1}{2}\left(2\pi r\right)=n\pi r$
Path $2$ $=$ $n\pi r$
So paths $1$ and $2$ will always be equal.