Answer:
The straight line distance from your starting point, after driving 14 miles east and then 6 miles north, is approximately 15.2 miles.
Explanation:
To understand how the straight-line distance is calculated in this scenario, we use the concept of a right-angled triangle. Here's a breakdown of the problem:
1. **Initial Scenario**:
- You start at a point and drive east for 14 miles.
- Then, you turn left (which means you turn north if you were initially heading east) and drive another 6 miles.
2. **Right-Angled Triangle Formation**:
- The path you've traveled forms a right-angled triangle.
- The first leg (when you drove east for 14 miles) forms one side of the triangle.
- The second leg (when you drove north for 6 miles) forms the other side of the triangle.
- The straight line from your starting point to your final position forms the hypotenuse of the triangle.
3. **Pythagoras' Theorem**:
- In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
- This is expressed as
, where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
4. **Applying the Theorem**:
- In your case, the eastward drive (14 miles) is \( a \), and the northward drive (6 miles) is \( b \).
- So, according to Pythagoras' theorem, the straight-line distance (hypotenuse, c can be calculated as:
![\[ c = √(a^2 + b^2) = √(14^2 + 6^2) \]](https://img.qammunity.org/2024/formulas/mathematics/college/4sag0qm7eg366ibxsk1tijreoqvvu6ho1w.png)
5. Calculation:
- When calculated, this gives approximately 15.2 miles.
6. Result:
- Hence, the straight-line distance from your starting point is approximately 15.2 miles.
This is a common application of Pythagorean theorem in determining the direct distance between two points in a two-dimensional plane, assuming straight eastward and northward paths form a right angle.