Vivian is deep-sea diving with Gina, who is to the south of her. A fish emerges from a hole to the east of Gina. The fish swims 12 meters to Gina and then swims 16 meters to Vivian. The fish then swims back to its hole and disappears. How far did the fish travel in all?

We can solve this problem by interpreting it as a geometry question in a coordinate plane, treating the movements of the fish as a right triangle.

1. **Position the Characters:**
   - Assume Vivian is at point \( V \), and Gina is directly south of her, at point \( G \). This means the line connecting \( V \) and \( G \) is vertical.
   - The fish emerges from a hole that is east of Gina, so the hole is at point \( H \), east of point \( G \), forming a right angle between the two.

2. **Label the Distances:**
   - The fish swims 12 meters from its hole \( H \) to Gina \( G \), which is the horizontal leg of a right triangle.
   - Then it swims 16 meters from Gina \( G \) to Vivian \( V \), which is the vertical leg of the right triangle.

3. **Use the Pythagorean Theorem:**
   The total distance from the hole \( H \) to Vivian \( V \) can be calculated using the Pythagorean theorem:
   \[
   HV = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \, \text{meters}
   \]
   
4. **Calculate the Total Distance:**
   The fish swims:
   - 12 meters from the hole \( H \) to Gina \( G \),
   - 16 meters from Gina \( G \) to Vivian \( V \),
   - 20 meters from Vivian \( V \) back to the hole \( H \).

   So, the total distance traveled by the fish is:
   \[
   12 + 16 + 20 = 48 \, \text{meters}
   \]

Thus, the fish traveled **48 meters** in total.

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Would you like more details or have any questions? Here are five related questions:

1. How would the total distance change if the fish swam different distances to Gina and Vivian?
2. Can we use other methods to find the distance traveled without the Pythagorean theorem?
3. How would the scenario change if Gina were to the west of Vivian instead of the south?
4. How could you represent this problem graphically using coordinate geometry?
5. What would happen if the fish took a different path back to the hole?

**Tip:** When dealing with right triangles, always check if the Pythagorean theorem can simplify the calculation of distances!

Solve a problem involving a fish traveling between two divers, Vivian and Gina, using the Pythagorean theorem. Learn how to calculate the total distance traveled, involving geometry concepts of right triangles and coordinate geometry. This is suitable for students in Grades 6-8.

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