Answer:
(b) 15, 36, 39
Explanation:
You want to know which sets of segment lengths could form a right triangle.
Right triangle
For segments to form a right triangle, the lengths must satisfy the triangle inequality and the Pythagorean theorem. The latter condition can often be determined by looking at the reduced ratios of the lengths, and comparing to known Pythagorean triples.
15, 30, 35
The ratio of lengths is 3:6:7. There are no integer side lengths that will form a right triangle when one of them is double another. Not a right triangle.
15, 36, 39
The ratio lengths is 5:12:13. These numbers are a common Pythagorean triple, so will form a right triangle.
15, 20, 29
The ratio of the smaller two numbers is 3:4. In order for these to form a right triangle, they must be part of a triangle with ratios 3:4:5. That would be 15, 20, 25. The side length 29 is too long for a right triangle. Not a right triangle.
5, 15, 30
The two short sides are too short to reach the ends of the long side. These lengths will not form a triangle.
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Additional comment
The table in the attachment computes a²+b²-c². When that value is 0, the Pythagorean theorem is satisfied, and the triangle is a right triangle. Positive numbers indicate an acute triangle; negative numbers indicate an obtuse triangle.
The Pythagorean triples that came into play in this answer are {3, 4, 5} and {5, 12, 13}. Other triples commonly seen are {7, 24, 25} and {8, 15, 17}.