Beginning in Town A, Biker Bob rode his bike 10 miles west, 3 miles north, 5 miles east, and then 9 miles north to Twon B. How far apart are Twon A and Twon B? (Assume perfectly flat terrain.)

Answer

To find the distance between Town A and Town B after Biker Bob's journey, we can break down the problem step by step.

### Step 1: Understand the Journey
Biker Bob starts at Town A and rides:
- 10 miles west
- 3 miles north
- 5 miles east
- 9 miles north

### Step 2: Determine the Final Position
Let's analyze Bob's movements in terms of coordinates, starting from Town A at the origin (0, 0).

1. **Move 10 miles west**:
- New position: (0 - 10, 0) = (-10, 0)

2. **Move 3 miles north**:
- New position: (-10, 0 + 3) = (-10, 3)

3. **Move 5 miles east**:
- New position: (-10 + 5, 3) = (-5, 3)

4. **Move 9 miles north**:
- New position: (-5, 3 + 9) = (-5, 12)

### Step 3: Calculate the Distance Between Town A and Town B
Now, we need to find the distance between Town A (0, 0) and Town B (-5, 12). We can use the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Where:
- \( (x_1, y_1) = (0, 0) \) (coordinates of Town A)
- \( (x_2, y_2) = (-5, 12) \) (coordinates of Town B)

Substituting the values:

\[ d = \sqrt{((-5) - 0)^2 + (12 - 0)^2} \]
\[ d = \sqrt{(-5)^2 + (12)^2} \]
\[ d = \sqrt{25 + 144} \]
\[ d = \sqrt{169} \]
\[ d = 13 \text{ miles} \]

### Conclusion
The distance between Town A and Town B is **13 miles**.