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

Hint: Some of the thing we need to know before solving this question:
Half of the circle is known as a semi-circle. Therefore, the area of the semi-circle will be half of the area of the circle.
Area of a circle is $$\pi r^2$$ then, area of a semicircle is $${\displaystyle \frac{\pi r^2}{2}}$$ where $$r$$ is the radius of circle or radius of semi-circle. Perimeter of a semicircle is given by, $$\pi r+2r$$ where $$r$$ is the radius of the semi-circle.

Complete step by step answer:
The given diagram consists of one large semi-circle and four small semi-circles. Thus, we have two paths to reach B from A.
It is given that the path $$1$$ from A to B is the large semi-circle that is the arc length of that semi-circle. That is the perimeter of a semi-circle minus its diameter, since this path doesn’t include the diameter. And it is given by $$\pi r+2r-d$$ where $$r$$ is the radius of the semi-circle and $$d$$ is the diameter of the semi-circle.
We know that diameter is nothing but twice the radius that is, $$d=2r$$ . Thus, we get, $$\pi r+2r+2r$$ .
On simplifying this we get, $$\pi r$$ . Thus, Path $$1=$$ $$\pi r$$ .
Let’s calculate the path $$2$$ . It is given that path $$2$$ is the way along the small four semi-circles. That is nothing but the perimeter of those four semicircles minus their diameters, since the path doesn’t cover its diameters. Since all four semi-circles are equal, we get, $$4\times (\pi r)$$ . Thus, Path $$2=$$ $$4\pi r$$ .
On comparing both parts we get, Path $$1<$$ Path $$2$$ . Since, $$\pi r<4\pi$$ is obvious.
Therefore, Path $$1$$ is smaller than Path $$2$$ .
Let us see the options, option (a) Path $$1$$ is longer than path $$2$$ this cannot be a correct answer since Path $$1$$ is smaller than Path $$2$$ .
Option (b) Path $$1$$ is of the same length as path $$2$$ , also this cannot be the correct answer since Path $$1$$ is smaller than Path $$2$$ .
Option (c) Path $$1$$ is shorter than path $$2$$ , this is the correct answer since Path $$1$$ is smaller than Path $$2$$ .
Option (d) Path $$1$$ is of the same length as path $$2$$ , only if the number of semi-circles is not more than $$4$$ , this cannot be the correct answer since Path $$1$$ is smaller than Path $$2$$ .

So, the correct answer is “Option C”.

Note: Since the path is from A to B, we cannot walk through the diameter of the semicircle clearly, they said that we need to walk through the arc. Thus, the path is the perimeter of the semi-circle except the diameter thus we neglected the diameter from the formula. Same idea is applied for those four semicircles.