Final answer:
After applying the Pythagorean theorem to each set of lengths, it's determined that option A. 4, 10, 16 does not satisfy the theorem and thus cannot represent the sides of a right triangle.
Step-by-step explanation:
The question asks which set of three numbers cannot represent the lengths of the sides of a right triangle. To determine this, we use the Pythagorean theorem which states that in a right triangle, the sum of the squares of the two shorter sides (legs) must equal the square of the longest side (hypotenuse). The formula for the Pythagorean theorem is a² + b² = c², where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
Using this theorem, we'll test each option:
- A. 4, 10, 16: 4² + 10² = 16², which simplifies to 16 + 100 = 256. This is not true since 116 does not equal 256.
- B. 5, 12, 13: 5² + 12² = 13², which simplifies to 25 + 144 = 169. This is true.
- C. 7, 24, 25: 7² + 24² = 25², which simplifies to 49 + 576 = 625. This is true.
- D. 8, 15, 17: 8² + 15² = 17², which simplifies to 64 + 225 = 289. This is true.
Therefore, the measures which cannot be the lengths of the sides of a right triangle is option A. 4, 10, 16.