To determine the total distance Jane’s trainer bikes using the Pythagorean Theorem, we need to consider the right triangle formed by her movements southward and westward.

Step 1: Identify the Right Triangle

• Let’s say Jane bikes A miles south and B miles west.
• The total distance she travels in a straight line back to her starting point (hypotenuse, C) can be found using the Pythagorean Theorem:
  $C^2 = A^2 + B^2$ where:
  • $A$ is the southward distance,
  • $B$ is the westward distance,
  • $C$ is the straight-line distance.

Step 2: Calculate the Distance

If we know the values for $A$ and $B$, we substitute them into the formula:

$C = \sqrt{A^2 + B^2}$

For example, if Jane bikes 3 miles south and 4 miles west: $C = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ miles}$

So, Jane's trainer bike distance in a straight line would be 5 miles.

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Do you need details or have any questions?

Here are some related questions to explore:

1. If Jane bikes 5 miles south and 12 miles west, what is the total distance back?
2. How would the formula change if she took a different diagonal route?
3. Can the Pythagorean Theorem be used in 3D movement (upward/downward paths)?
4. What if Jane's route forms a non-right triangle—how would we solve it?
5. How can this be applied to real-life navigation using maps?

Tip: The Pythagorean Theorem only works with right triangles—for other triangles, use the law of cosines! 🚴‍♀️📐