in says she has memorized the lengths of a few right triangles, for example, 3,4 , and She is trying to compile a list of several right triangles but needs your help. select all the lengths of triangles that are right. (Lesson 3 -14) \( \begin{array}{ll}\text { A) } 1,1, \sqrt{2} \\ \text { B) } 2,4,6 \\ \text { (C) } 5,12,13 \\ \text { D) } 8,9,64 \\ \text { (E) } 7,19,22\end{array} \)

To determine which sets of lengths can form right triangles, we can apply the Pythagorean theorem, which states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the equation \(a^2 + b^2 = c^2\) must hold true. So, let’s dive into each option: A) \(1, 1, \sqrt{2}\): \(1^2 + 1^2 = 2\) and \((\sqrt{2})^2 = 2\) ✓ B) \(2, 4, 6\): \(2^2 + 4^2 = 4 + 16 = 20\) and \(6^2 = 36\) ✗ C) \(5, 12, 13\): \(5^2 + 12^2 = 25 + 144 = 169\) and \(13^2 = 169\) ✓ D) \(8, 9, 64\): \(8^2 + 9^2 = 64 + 81 = 145\) and \(64^2 = 4096\) ✗ E) \(7, 19, 22\): \(7^2 + 19^2 = 49 + 361 = 410\) and \(22^2 = 484\) ✗ So, the lengths that can form right triangles are: **A) \(1, 1, \sqrt{2}\)** and **C) \(5, 12, 13\)**. Want to dive deeper into triangles? Right triangles have such storied histories in different cultures, especially in ancient Greece with Pythagoras! He and his followers made significant contributions to understanding these triangles, influencing mathematics for centuries. For modern-day applications, right triangles are everywhere! From construction to navigation, knowledge of right triangles helps ensure structures are built accurately and calculating distances becomes effortless due to concepts like trigonometric ratios. So, next time you pass a building site or look at a map, you can think of that trusty right triangle!