When Ted earned his driver's licence, he wanted his first solo drive to be a friend's house. Previously. Ted had always biked to his friend's hours and was able to cut through the yards of neighbors to travel in a straight line. In his car, however, Ted travels a longer distance as he follows the streets. As a result, he travels 6 miles east, 6 miles south, and 2 more miles east by car. How much shorter, in miles, is Ted's bike route than his car route ?

Answer

To solve the problem, we need to find the difference in distance between Ted's car route and his bike route.

### Step-by-step Solution:

1. **Calculate the Total Distance of Ted's Car Route:**
- Ted travels 6 miles east, then 6 miles south, and finally 2 more miles east.
- Total distance by car = 6 miles (east) + 6 miles (south) + 2 miles (east) = 14 miles.

2. **Visualize Ted's Bike Route:**
- When biking, Ted travels in a straight line from the starting point to his friend's house.
- The path can be visualized as a right triangle where:
- One leg (east) is 6 miles + 2 miles = 8 miles (total east distance).
- The other leg (south) is 6 miles.

3. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem, for a right triangle:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the hypotenuse (the straight-line distance), and \(a\) and \(b\) are the other two sides.
- Here, \(a = 8\) miles (east) and \(b = 6\) miles (south).
- Therefore, we calculate:
\[ c^2 = 8^2 + 6^2 = 64 + 36 = 100 \]
- Taking the square root gives:
\[ c = \sqrt{100} = 10 \text{ miles} \]

4. **Calculate the Difference in Distance:**
- Now, we find out how much shorter the bike route is compared to the car route:
\[ \text{Difference} = \text{Car Route} - \text{Bike Route} = 14 \text{ miles} - 10 \text{ miles} = 4 \text{ miles} \]

### Final Answer:
Ted's bike route is **4 miles shorter** than his car route.