Two hikers started at the same location. One traveled 2 miles east and then 1 mile north. The other traveled 1 mile west and then 3 miles south. At the end of their hikes, how many miles apart were the two hikers?

Answer

To solve the problem of finding the distance between the two hikers after they have completed their journeys, we can follow these steps:

### Step 1: Determine the final positions of each hiker.
- **Hiker 1** travels 2 miles east and then 1 mile north.
- Starting point: (0, 0)
- After moving 2 miles east: (2, 0)
- After moving 1 mile north: (2, 1)

- **Hiker 2** travels 1 mile west and then 3 miles south.
- Starting point: (0, 0)
- After moving 1 mile west: (-1, 0)
- After moving 3 miles south: (-1, -3)

### Step 2: Identify the coordinates of both hikers.
- Hiker 1's final coordinates: (2, 1)
- Hiker 2's final coordinates: (-1, -3)

### Step 3: Use the distance formula to find the distance between the two hikers.
The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the hikers:
- \( x_1 = 2, y_1 = 1 \)
- \( x_2 = -1, y_2 = -3 \)

### Step 4: Plug in the values into the distance formula.
\[ d = \sqrt{((-1) - 2)^2 + ((-3) - 1)^2} \]
\[ d = \sqrt{(-3)^2 + (-4)^2} \]
\[ d = \sqrt{9 + 16} \]
\[ d = \sqrt{25} \]
\[ d = 5 \]

### Conclusion:
The distance between the two hikers at the end of their hikes is **5 miles**.

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