Select 3 side lengths that can form a right triangle. 12 centimeters 14 centimeters 16 centimeters 30 centimeters 32 centimeters 34 centimeters

To determine which sets of side lengths can form a right triangle, we can use the Pythagorean theorem. According to this theorem, for three sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the following condition must hold: \[ a^2 + b^2 = c^2 \] Let's check both sets of side lengths: ### Set 1: 12 cm, 14 cm, 16 cm 1. Identify the longest side: \(c = 16\) cm, \(a = 12\) cm, \(b = 14\) cm. 2. Calculate \(a^2 + b^2\) and \(c^2\): - \(a^2 + b^2 = 12^2 + 14^2 = 144 + 196 = 340\) - \(c^2 = 16^2 = 256\) Since \(340 \neq 256\), this set does not form a right triangle. ### Set 2: 30 cm, 32 cm, 34 cm 1. Identify the longest side: \(c = 34\) cm, \(a = 30\) cm, \(b = 32\) cm. 2. Calculate \(a^2 + b^2\) and \(c^2\): - \(a^2 + b^2 = 30^2 + 32^2 = 900 + 1024 = 1924\) - \(c^2 = 34^2 = 1156\) Since \(1924 \neq 1156\), this set does not form a right triangle. ### Conclusion Neither of the sets of side lengths can form a right triangle.