The side lengths of four triangles are shown in the table. Select all the triangles that are right triangles.
Triangle	Side Lengths
|	610
810
1
||	7	7	13
|||	313
413
513
IV	5	6	11


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.

The side lengths of four triangles are shown in the table. Select all the triangles that are right triangles.
Triangle	Side Lengths
| 610 810 1 |
|| 7 7 13 |||
||| 313 413 513 IV
| 5 6 11 |

Explore how to determine whether a triangle is a right triangle by applying the Pythagorean Theorem. This step-by-step guide evaluates four triangles with side lengths such as 610, 810, and 1, and shows why none qualify as right triangles. Ideal for students in Grades 8-10.

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