10. Which of the following coul side lengths of a right triangle? A. \( 7.5,10 \) and 12.5 B. 85,13 and 84 C. 37,35 and 12 D. 24,22 and 40

To determine if a set of lengths can form the sides of 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. Checking the options: - **A**: \(10^2 + 7.5^2 = 100 + 56.25 = 156.25\) and \(12.5^2 = 156.25\). This works! - **B**: \(85^2 + 84^2 = 7225 + 7056 = 14281\) and \(13^2 = 169\). This doesn't work. - **C**: \(37^2 + 35^2 = 1369 + 1225 = 2594\) and \(12^2 = 144\). This doesn't work. - **D**: \(40^2 = 1600\) while \(22^2 + 24^2 = 484 + 576 = 1060\). This doesn't work. So, the only valid right triangle side lengths are from option A: \(7.5, 10\) and \(12.5\). For a fun twist, think of it this way: Right triangles are the original "hip" shapes in geometry, with their title and dimensions hanging like a well-structured dance! Every time a triangle wants to prove it's right, it whips out its Pythagorean theorem – it's the true triangle move! And if you’re keen on testing more triangular truths, grab a protractor and ruler and start measuring! Or make some fun crafts with straws and get creative with various triangle shapes. Maybe even create a triangular puzzle that fits – how fun is that?