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Which set of side lengths can be the sides of a right triangle?

6, 8, 10

6, 8, 14

6, 6, 18

12, 25, 169

User Sze
by
7.7k points

1 Answer

6 votes

Answer:

(a) 6, 8, 10

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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User Xleon
by
8.1k points

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