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Answer for 15 points

Answer for 15 points-example-1

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Answer:

6.1, 13.7, 8.2: Cannot be side lengths of a triangle

10, 15, 27: Cannot be side lengths of a triangle

15, 8, 14: Cannot be side lengths of a triangle

13, 12, 14: Can be side lengths of a triangle

Explanation:

According to the Triangle Inequality Theorem, three lengths are able to form a triangle if the sum of the squares of any two lengths is greater than the square of the third length.

Because there are three lengths, we have to repeat the process unless we find a violation of the theorem, as one violation automatically disqualifies three lengths from forming a triangle.

'll refer to the three lengths as a, b, and c for the sake of clarity.

Checking 6.1 (a), 13.7 (b), and 8.2 (c):

a^2 + b^2 > c^2

6.1^2 + 13.7^2 > 8.2^2

37.21 + 187.69 > 67.24

224.9 > 67.24

b^2 + c^2 > a^2

13.7^2 + 8.2^2 > 6.1^2

187.69 + 67.24 > 37.21

254.93 > 37.21

c^2 + a^2 > b^2

8.2^2 + 6.1^2 > 13.7^2

67.24 + 37.21 > 187.69

104.56 < 187.69.

Because the sum of the squares of 8.2 and 6.2 is less than the square of 13.7, these three lengths can't form a triangle.

Checking 10 (a), 15 (b), and 27 (c):

a^2 + b^2 > c^2

10^2 + 15^2 > 27^2

100 + 225 > 729

325 < 729

Because the sum of the squares of 10 and 15 is less than the square of 27, these three lengths can't form a triangle:

Checking 15 (a), 8 (b), and 14 (c):

a^2 + b^2 > c^2

15^2 + 8^2 > 14^2

225 + 64 > 196

289 > 196

b^2 + c^2 > a^2

8^2 + 14^2 > 15^2

64 + 196 > 225

260 > 225

c^2 + a^2 > b^2

14^2 + 15^2 > 8^2

196 + 225 > 64

421 < 64

Because the sum of the squares of 14 and 15 is less than the square of 8, these three lengths can't form a triangle.

Checking 13 (a), 12 (b), and 14 (c):

a^2 + b^2 > c^2

13^2 + 12^2 > 14^2

169 + 144 > 196

313 > 196

b^2 + c^2 > a^2

12^2 + 14^2 > 13^2

144 + 196 > 169

340 > 169

c^2 + a^2 > b^2

14^2 + 13^2 > 12^2

196 + 169 > 144

365 > 144

Because all three lengths satisfy the Triangle Inequality Theorem, 13, 12, and 14 can form a triangle.

User Juan Salcedo
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