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

Options

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

Answer

To solve the problem step by step, we will first visualize the scenario and then calculate the total distance Shyam walked.

### Step 1: Visualize the Positions
- Shyam's house (H), office (O), and gym (G) form an equilateral triangle where each side is 4 km.
- We can place these points on a coordinate system for clarity:
- Let H be at (0, 0).
- Let O be at (4, 0).
- To find the coordinates of G, we can use the properties of an equilateral triangle. The height (h) can be calculated using the formula \( h = \sqrt{s^2 - (s/2)^2} \), where \( s \) is the side length (4 km):
\[ h = \sqrt{4^2 - (4/2)^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \approx 3.46 \]
- Thus, G can be at (2, 2√3).

### Step 2: Shyam's First Walk
- Shyam starts walking from G (2, 2√3) towards a point directly east of O (4, 0).
- Since he walks parallel to the line connecting H and O, he will walk horizontally to the right until he reaches the line y = 0.
- The distance he covers in this step is the horizontal distance from G to the point directly east of O:
\[ \text{Distance} = 4 - 2 = 2 \text{ km} \]

### Step 3: Shyam's Second Walk
- After reaching the point directly east of O, Shyam reverses direction and walks directly south to reach a point directly south of O.
- The coordinates of the point directly south of O (4, 0) would be (4, -y), where y is the vertical distance from O to the line y = 0.
- The distance he needs to cover to go from (4, 0) to (4, -2√3) is:
\[ \text{Distance} = 2√3 \text{ km} \]

### Step 4: Calculate Total Distance
- Now, we can sum the distances from both steps:
\[ \text{Total Distance} = 2 \text{ km} + 2√3 \text{ km} \]
- Approximating \( 2√3 \) (where \( √3 \approx 1.732 \)):
\[ 2√3 \approx 2 \times 1.732 \approx 3.464 \text{ km} \]
- Thus, the total distance:
\[ \text{Total Distance} \approx 2 + 3.464 = 5.464 \text{ km} \]

### Final Answer
The total distance walked by Shyam is approximately **5.464 km**.