Question

Shyam's house, his office and his gym are all equidistant from each other. The distance between any 2 of them is 4 km. Shyam starts walking from his gym in a direction parallel to the road connecting his office and his house and stops when he reaches a point directly east of his office. He then reverses direction and walks till he reaches a point directly south of his office. The total distance walked by Shyam is

• A. 9 km
• B. 6 km
• C. 16 km
• D. 12 km

Hint:

Draw the equilateral triangle and track each straight-line leg using symmetry about the east-west line through the office.

Answer 1

The Correct Option is D

Step 1: Set up the equilateral triangle.
Office, House and Gym form an equilateral triangle with each side equal to 4 km, since all three places are equidistant from each other. For a point directly east of the office and a point directly south of the office to both lie on Shyam's straight-line path, House and Gym must sit symmetrically on either side of the east-west line through the Office, that is, the Office-House road and the Office-Gym road make equal angles with due east, one swinging north and one swinging south.

Step 2: Find the distance to the point east of the office.
Shyam starts at the Gym and walks in a direction parallel to the Office-House road. Because of the symmetry described above, walking from the Gym parallel to Office-House for a distance equal to one side of the triangle (4 km) lands him exactly on the east-west line through the Office, that is, at the point directly east of the Office. This first leg is therefore 4 km.

Step 3: Find the distance to the point south of the office.
He now reverses direction and retraces the same straight line. Walking back 4 km brings him exactly to the Gym again. By the same symmetry, walking a further 4 km beyond the Gym along this same line brings him to a point exactly 4 km due south of the Office. So this return leg covers 4 km (back to the Gym) plus 4 km (Gym to the south point) = 8 km.

Final Answer:
Total distance walked = 4 km (Gym to east point) + 8 km (back to Gym and on to the south point) = 12 km. \[ \boxed{12 \text{ km}} \]