This is a right triangle, and we are tasked with finding the length of the hypotenuse $c$. Using the Pythagorean theorem, the formula is:

$c^2 = a^2 + b^2$

Here:

• $a = 12 \, \text{km}$
• $b = 16 \, \text{km}$

Step-by-step Calculation:

1. Substitute the values into the formula: $c^2 = 12^2 + 16^2$

2. Compute the squares of 12 and 16: $c^2 = 144 + 256$

3. Add the results: $c^2 = 400$

4. Take the square root of both sides: $c = \sqrt{400} = 20 \, \text{km}$

Final Answer:

The length of the hypotenuse is: $c = 20 \, \text{km}$

Would you like me to explain this further or assist with any additional questions?

Here are 5 related questions to expand your understanding:

1. How is the Pythagorean theorem derived?
2. What happens if one of the legs of the triangle is unknown—how can you solve for it?
3. How would you apply the Pythagorean theorem to find diagonal lengths in 3D shapes like cubes?
4. What are real-world applications of the Pythagorean theorem?
5. How can you determine if a triangle is a right triangle using side lengths?

Tip: Always double-check your work by substituting the values back into the original equation to ensure your solution is correct!