Try this beautiful Geometry problem from PRMO, 2019 based on Direction & Angles.

## Direction & Angles | PRMO | Problem-4

An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160Â° turn to the right and walks 4 more feet. It then makes another 160Â° turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is the distance in feet it would have walked?

- $20$
- $36$
- $13$

**Key Concepts**

Geometry

Co-ordinate geometry

Trigonometry

## Check the Answer

But try the problem first…

Answer:\(36\)

PRMO-2018, Problem 13

Pre College Mathematics

## Try with Hints

First hint

According to the problem we draw the figure and try to solve using co-ordinate method…

Can you now finish the problem ……….

Second Hint

Let \(A_0(0,0)\)

Therefore \(A_1(4 cos 0,4sin 0)\)

\(A_2(4cos 0+4 cos 160, 4 sin 0+4 sin 160)\)

\(\Rightarrow A_n =(0,0)\)

and \(4(cos 0+cos 160 +…+sin 160(n-1)=0\)

\(\Rightarrow n =9\)

Can you finish the problem……..

Final Step

Therefore distance covered =\(4 \times 9\)=\(36\)

## Other useful links

- https://www.cheenta.com/problem-based-on-triangle-prmo-2012-problem-7/
- https://www.youtube.com/watch?v=TRf4TgrOIpA

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