The houses of A and B face each other on a road going north-south, A's being on the western side. A comes out of his house, turns left, travels 5 km, turns right, travels 5 km to the front of D's house. B does exactly the same and reaches the front of C's house. In this context, which one of the following statements is correct?

Let's analyze the movements and positions:

1. Initial Setup:

   • Road goes North-South.
   • A's house is on the Western side.
   • B's house is on the Eastern side, facing A's house.
   • Let's place A's house at the origin (0,0).
   • Since B's house faces A's house on the Eastern side, B's house will be at (w,0), where 'w' is the width of the road.

2. A's Movement to D's House:

   • A comes out of his house (from (0,0), facing East).
   • Turns left (now facing North).
   • Travels 5 km (reaches (0, 5)).
   • Turns right (now facing East).
   • Travels 5 km to D's house (reaches (5, 5)).
   • So, D's house is at (5, 5).

3. B's Movement to C's House:

   • B does exactly the same. B comes out of his house (from (w,0), facing West).
   • Turns left (now facing South).
   • Travels 5 km (reaches (w, -5)).
   • Turns right (now facing West).
   • Travels 5 km to C's house (reaches (w - 5, -5)).
   • So, C's house is at (w - 5, -5).

4. Calculate Distance between C and D:

   • D = (5, 5)
   • C = (w - 5, -5)
   • Distance = sqrt[ (x2 - x1)^2 + (y2 - y1)^2 ]
   • Distance = sqrt[ ( (w - 5) - 5 )^2 + ( -5 - 5 )^2 ]
   • Distance = sqrt[ (w - 10)^2 + (-10)^2 ]
   • Distance = sqrt[ (w - 10)^2 + 100 ]

Now let's evaluate each option:

A) C and D live on the same street * D is at (5, 5) and C is at (w - 5, -5). * For them to be on the same street, their x-coordinates or y-coordinates would need to be the same. * Their y-coordinates (5 and -5) are different. * Their x-coordinates (5 and w-5) are only the same if w = 10. This is not generally true for any road width. * Therefore, statement A is incorrect.

B) C's house faces south * B reaches the front of C's house after traveling 5 km West. This means B is facing West when he arrives at C's house. * If B is facing West at the front of C's house, then C's house must be facing East. * Therefore, statement B is incorrect.

C) The houses of C and D are less than 20 km apart * We need to check if sqrt[ (w - 10)^2 + 100 ] 0). * So, the distance is less than 20 km if 0 < w < 27.32 km. * While the problem does not specify 'w', in such questions, it's usually implied that 'w' falls within a realistic range for a "road" where houses face each other. A road width up to approximately 27 km is very wide but still conceivable (e.g., a very wide boulevard, or a highway with a large median). The minimum distance (when w=10) is 10 km, which is less than 20 km. Given that options A and B are definitively incorrect, this option is the most plausible and considered correct under the assumption of a practical road width.

D) None of the above * Since C is correct under reasonable assumptions, this option is incorrect.

The final answer is C.