Question

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

• A. Path I is longer than Path II
• B. Path I of the same length of Path II
• C. Path I is shorter than Path II
• D. Path I is of the same length as Path II. Only if the number of semi circles is not more than $4$

Answer 1

The correct option is A Path I is longer than Path II

Let the length of $AB$ be $d$

So the radius of larger semi circle is $\frac{d}{2}$ and the radius of smaller semi circle is $\frac{d}{8}$

So the length of $AB$ through path 1 is $\pi \times \frac{d}{2}=\frac{\pi d}{2}$

So the length of $AB$ through path 2 is $4\times \pi \times \frac{d}{8}=\frac{\pi d}{2}$

So the both paths are of same length