Final answer:
Using the Pythagorean theorem, it is shown that option (b), with sides 6 in, 10 in, and 4 in, does not satisfy the condition a² + b² = c², and is therefore not a right triangle.
Step-by-step explanation:
To determine which set of sides does not form a right triangle, we use the Pythagorean theorem, which states for any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is expressed as a² + b² = c².
- (a) 27 cm, 36 cm, 45 cm: 27² + 36² = 45², or 729 + 1296 = 2025, which is true.
- (b) 6 in, 10 in, 4 in: 6² + 4² != 10², or 36 + 16 != 100, which is false. Therefore, this is not a right triangle.
- (c) 3 cm, 4 cm, 5 cm: 3² + 4² = 5², or 9 + 16 = 25, which is true.
- (d) 6 ft, 8 ft, 12 ft: 6² + 8² != 12², or 36 + 64 != 144, which is false. However, since we are looking for a single triangle that is not right, and we've already found one, we now know option (b) is the answer.
Thus, option (b) is the one that does not represent a right triangle.