To determine if each triangle is a right triangle, we can use the Pythagorean Theorem, which states that for a right triangle with sides $a$, $b$, and hypotenuse $c$, the following must hold true:

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

Let’s evaluate each triangle:

Triangle I: Side lengths 610, 810, and 1

• Here, the largest side is $c = 810$, and $a = 610$, $b = 1$.
• Applying the Pythagorean theorem: $610^2 + 1^2 = 372100 + 1 = 372101$ $810^2 = 656100$ Clearly, $372101 \neq 656100$, so Triangle I is not a right triangle.

Triangle II: Side lengths 7, 7, and 13

• The largest side is $c = 13$, and $a = 7$, $b = 7$.
• Applying the Pythagorean theorem: $7^2 + 7^2 = 49 + 49 = 98$ $13^2 = 169$ Since $98 \neq 169$, Triangle II is not a right triangle.

Triangle III: Side lengths 313, 413, and 513

• The largest side is $c = 513$, and $a = 313$, $b = 413$.
• Applying the Pythagorean theorem: $313^2 + 413^2 = 97969 + 170569 = 268538$ $513^2 = 263169$ Since $268538 \neq 263169$, Triangle III is not a right triangle.

Triangle IV: Side lengths 5, 6, and 11

• The largest side is $c = 11$, and $a = 5$, $b = 6$.
• Applying the Pythagorean theorem: $5^2 + 6^2 = 25 + 36 = 61$ $11^2 = 121$ Since $61 \neq 121$, Triangle IV is not a right triangle.

Conclusion:

None of the triangles listed are right triangles.

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. What is the definition of an acute triangle?
2. How can we determine if a triangle is obtuse?
3. How does the Law of Cosines relate to the Pythagorean theorem?
4. Can a triangle with equal side lengths be a right triangle?
5. What are the properties of a scalene triangle?

Tip: When dealing with triangle problems, always check whether the largest side could be the hypotenuse by testing it with the Pythagorean Theorem.