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To determine if each triangle is a right triangle, we can use the **Pythagorean theorem**. For a triangle with side lengths \( a \), \( b \), and hypotenuse \( c \), it will be a right triangle if:

\[
a^2 + b^2 = c^2
\]

I'll check each triangle in the image.

### Triangle 1
- Side lengths: \( a = 5 \), \( b = 2 \), and hypotenuse \( c = \sqrt{21} \).
- Calculate \( a^2 + b^2 \):
  \[
  5^2 + 2^2 = 25 + 4 = 29
  \]
- Calculate \( c^2 \):
  \[
  (\sqrt{21})^2 = 21
  \]
- Since \( 29 \neq 21 \), this is **not a right triangle**.

### Triangle 2
- Side lengths: \( a = \frac{3}{8} \), \( b = \frac{6}{8} \), and hypotenuse \( c = \frac{5}{8} \).
- Calculate \( a^2 + b^2 \):
  \[
  \left(\frac{3}{8}\right)^2 + \left(\frac{6}{8}\right)^2 = \frac{9}{64} + \frac{36}{64} = \frac{45}{64}
  \]
- Calculate \( c^2 \):
  \[
  \left(\frac{5}{8}\right)^2 = \frac{25}{64}
  \]
- Since \( \frac{45}{64} \neq \frac{25}{64} \), this is **not a right triangle**.

### Triangle 3
- Side lengths: \( a = 0.8 \), \( b = 0.6 \), and hypotenuse \( c = 1 \).
- Calculate \( a^2 + b^2 \):
  \[
  (0.8)^2 + (0.6)^2 = 0.64 + 0.36 = 1
  \]
- Calculate \( c^2 \):
  \[
  1^2 = 1
  \]
- Since \( 1 = 1 \), this **is a right triangle**.

### Summary
- **Triangle 1**: Not a right triangle
- **Triangle 2**: Not a right triangle
- **Triangle 3**: Right triangle

Would you like a deeper explanation of the steps? 

Here are five related questions to deepen your understanding:
1. How does the Pythagorean theorem determine if a triangle is a right triangle?
2. What would happen if the sides of a triangle only approximately matched the Pythagorean theorem?
3. Can non-right triangles have a relationship similar to the Pythagorean theorem?
4. How can side lengths be scaled to keep a triangle right?
5. What other methods can confirm if a triangle is right?

**Tip:** When verifying right triangles, always check if the longest side can be the hypotenuse in the Pythagorean theorem.

Indicate whether each triangle is a right triangle.

Learn how to determine if a triangle is a right triangle by using the Pythagorean theorem. This problem explores geometric concepts and applies the formula a^2 + b^2 = c^2 to identify right triangles, suitable for students in Grades 6-8.

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