The image shows four right triangles, each with two legs given, and the hypotenuse labeled as "c." To find the hypotenuse $c$, we can use the Pythagorean theorem, which states:

$c = \sqrt{a^2 + b^2}$

where $a$ and $b$ are the lengths of the two legs of the triangle, and $c$ is the hypotenuse.

1. Triangle 1:

• Legs: 3.4 cm and 4.5 cm $c = \sqrt{(3.4)^2 + (4.5)^2} = \sqrt{11.56 + 20.25} = \sqrt{31.81} \approx 5.64 \, \text{cm}$

2. Triangle 2:

• Legs: 16 in and 12.8 in $c = \sqrt{(16)^2 + (12.8)^2} = \sqrt{256 + 163.84} = \sqrt{419.84} \approx 20.49 \, \text{in}$

3. Triangle 3:

• Legs: 12.9 mi and 12.6 mi $c = \sqrt{(12.9)^2 + (12.6)^2} = \sqrt{166.41 + 158.76} = \sqrt{325.17} \approx 18.03 \, \text{mi}$

4. Triangle 4:

• Legs: 10.2 mi and 8.7 mi $c = \sqrt{(10.2)^2 + (8.7)^2} = \sqrt{104.04 + 75.69} = \sqrt{179.73} \approx 13.41 \, \text{mi}$

Summary:

1. Hypotenuse for Triangle 1: $\approx 5.64 \, \text{cm}$
2. Hypotenuse for Triangle 2: $\approx 20.49 \, \text{in}$
3. Hypotenuse for Triangle 3: $\approx 18.03 \, \text{mi}$
4. Hypotenuse for Triangle 4: $\approx 13.41 \, \text{mi}$

Would you like more details on any step, or have any further questions?

Here are 5 follow-up questions:

1. What are the properties of a right triangle?
2. Can you derive the Pythagorean theorem from a geometrical perspective?
3. How does the unit of measurement affect the calculation?
4. What is the relationship between the Pythagorean theorem and trigonometric functions?
5. How would you find the area of each triangle?

Tip: Always ensure units are consistent when using the Pythagorean theorem for mixed units (like inches and feet).