Which of the following side lengths would make up a right triangle? \( 3,6,9 \) \( 7,24,25 \) \( 14.2,124,16.5 \) \( 1.5,1.5,3 \)

To determine if three side lengths can form a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. For the sets provided: 1. **\( 3, 6, 9 \)**: \( 9^2 = 81 \) and \( 3^2 + 6^2 = 9 + 36 = 45 \) (not a right triangle) 2. **\( 7, 24, 25 \)**: \( 25^2 = 625 \) and \( 7^2 + 24^2 = 49 + 576 = 625 \) (a right triangle) 3. **\( 14.2, 124, 16.5 \)**: \( 124^2 = 15376 \) and \( 14.2^2 + 16.5^2 = 201.64 + 272.25 = 473.89 \) (not a right triangle) 4. **\( 1.5, 1.5, 3 \)**: \( 3^2 = 9 \) and \( 1.5^2 + 1.5^2 = 2.25 + 2.25 = 4.5 \) (not a right triangle) So, only \( 7, 24, 25 \) can form a right triangle! Now, how about testing out your triangle knowledge with a classic in geometry? The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. It's a useful tool to rule out non-triangles before you even get to the right or acute angles! And don’t forget, if you want to dive deeper into the magic of triangles, consider checking out Euclid's "Elements." It’s like the celestial map of geometry, laying down not only the sides of triangles but also the angles and principles that govern their entire universe!