Answer:
(a) 6, 8, 10
Step-by-step explanation:
You want to know the side lengths that form a right triangle from among the choices offered.
Triangle
First, we can eliminate all choices where the side lengths cannot form a triangle. A triangle will only be formed if the sum of the shortest two lengths exceeds the longest.
6 + 8 > 10 . . . . true, forms a triangle
6 + 8 > 14 . . . . false, not a triangle
6 + 6 > 18 . . . . false, not a triangle
12 +25 > 169 . . . . false, not a triangle
Right triangle
Already we know there is only one reasonable answer choice:
6, 8, 10
We can check to see if these lengths form a right triangle. If they do, they will satisfy the Pythagorean relation:
a² +b² = c²
6² +8² = 10²
36 + 64 = 100 . . . . true
The side lengths 6, 8, 10 can be the sides of a right triangle.
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Additional comment
The sides 6, 8, 10, have the ratios 3:4:5. They are double the lengths of the primitive Pythagorean triple {3, 4, 5}. It can be worthwhile to remember a few such triples, as they appear often in problems involving triangles. Here are a few:
{3, 4, 5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}, {9, 40, 41}
The only triple that is an arithmetic sequence is 3, 4, 5. Another point worthy of note is that the sum of integer side lengths of a right triangle is always an even number.
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