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.