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.