To determine which of the given options represent the lengths of the sides of a right triangle, we can use the Pythagorean Theorem. This theorem states that for a right triangle with sides $a$, $b$, and hypotenuse $c$:

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

Let’s go through each option:

• A: 1, 1, and 2 $1^2 + 1^2 = 1 + 1 = 2 \neq 2^2 = 4$ This does not satisfy the Pythagorean theorem.

• B: 18, 24, and 30 $18^2 + 24^2 = 324 + 576 = 900 \quad \text{and} \quad 30^2 = 900$ This satisfies the Pythagorean theorem. This is a valid right triangle.

• C: 12, 15, and 19 $12^2 + 15^2 = 144 + 225 = 369 \neq 19^2 = 361$ This does not satisfy the Pythagorean theorem.

• D: 2, 2, and 4 $2^2 + 2^2 = 4 + 4 = 8 \neq 4^2 = 16$ This does not satisfy the Pythagorean theorem.

• E: 5, 12, and 13 $5^2 + 12^2 = 25 + 144 = 169 \quad \text{and} \quad 13^2 = 169$ This satisfies the Pythagorean theorem. This is a valid right triangle.

Correct answers:

• B: 18, 24, and 30
• E: 5, 12, and 13

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How can we generate Pythagorean triples systematically?
2. Can the Pythagorean theorem be applied to non-Euclidean geometries?
3. What are some real-world applications of the Pythagorean theorem?
4. How does the converse of the Pythagorean theorem work?
5. Are there any shortcuts for checking if three numbers form a right triangle?

Tip: The simplest form of a Pythagorean triple is $3, 4, 5$, and multiples of these also form right triangles.