‘A’ started from his house and walked 20 m towards East, where his friend ‘B’ joined him. They together walked 10 m in the same direction. Then ‘A’ turned left while ‘B’ turned right and travelled 2 m and 8 m respectively. Again ‘B’ turned’ left to travel 4 m followed by 5 m to his right to reach his office. ‘A’ turned right and travelled 12 m to reach his office. What is the shortest distance between the two offices?

The correct option is 17 m.

Let us break down the movements of both 'A' and 'B' step-by-step from their starting positions to their respective offices using a standard coordinate system. Let the starting point (A's house) be the origin (0, 0). Let the East direction be the positive x-axis, West be the negative x-axis, North be the positive y-axis, and South be the negative y-axis.

Step 1: Initial Walk
'A' starts at (0, 0) and walks 20 m towards East. He reaches point (20, 0), where his friend 'B' joins him.
They together walk another 10 m in the same direction (East).
They are now both at position:
P = (20 + 10, 0) = (30, 0).

Step 2: Movements of 'A' and 'B' from Point P(30, 0)
At point (30, 0), both are facing East.
- 'A' turns left: A left turn from facing East means 'A' now faces North. He travels 2 m.
So, 'A' moves to position:
A₁ = (30, 0 + 2) = (30, 2).
- 'B' turns right: A right turn from facing East means 'B' now faces South. He travels 8 m.
So, 'B' moves to position:
B₁ = (30, 0 - 8) = (30, -8).

Step 3: Further movements to reach their offices
- For 'A': From A₁(30, 2), facing North, 'A' turns right (which is East) and travels 12 m to reach his office.
So, A's office position (O_A) is:
O_A = (30 + 12, 2) = (42, 2).
- For 'B': From B₁(30, -8), facing South, 'B' turns left (which is East) to travel 4 m.
His position becomes:
B₂ = (30 + 4, -8) = (34, -8), now facing East.
From B₂(34, -8), facing East, 'B' turns right (which is South) and travels 5 m to reach his office.
So, B's office position (O_B) is:
O_B = (34, -8 - 5) = (34, -13).

Step 4: Calculate the shortest distance between the two offices
The two office coordinates are O_A(42, 2) and O_B(34, -13). We use the distance formula to find the shortest distance d:
$$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$$ Substitute the coordinates into the formula:
$$d=\sqrt{{(34-42)}^2+{(-13-2)}^2}$$ $$d=\sqrt{{(-8)}^2+{(-15)}^2}$$ $$d=\sqrt{64+225}$$ $$d=\sqrt{289}$$ $$d=17\text{ m}$$

Thus, the shortest distance between the two offices is 17 m.