Answer:
{6, 8, 10} is a set which represents the side length of a right triangle.
Step-by-step explanation:
In a right triangle:
![(Base)^{2} + (Perpendicular)^{2} = (Hypotenuse)^{2}](https://tex.z-dn.net/?f=%28Base%29%5E%7B2%7D%20%20%2B%20%28Perpendicular%29%5E%7B2%7D%20%20%20%3D%20%28Hypotenuse%29%5E%7B2%7D)
Now, in the given triplets:
(a) {4, 8, 12}
Here, ![(4)^{2} + (8)^{2} = 16 + 64 = 80\\\implies H = \sqrt{80} = 8.94](https://tex.z-dn.net/?f=%284%29%5E%7B2%7D%20%20%2B%20%288%29%5E%7B2%7D%20%20%20%3D%2016%20%2B%2064%20%20%3D%2080%5C%5C%5Cimplies%20H%20%3D%20%5Csqrt%7B80%7D%20%20%3D%20%208.94)
So, third side of the triangle 8.94 ≠ 12
Hence, {4, 8, 12} is NOT a triplet.
(b) {6, 8, 10}
Here, ![(6)^{2} + (8)^{2} = 36 + 64 = 100\\\implies H = \sqrt{100} = 10](https://tex.z-dn.net/?f=%286%29%5E%7B2%7D%20%20%2B%20%288%29%5E%7B2%7D%20%20%20%3D%2036%20%2B%2064%20%20%3D%20100%5C%5C%5Cimplies%20H%20%3D%20%5Csqrt%7B100%7D%20%20%3D%20%2010)
So, third side of the triangle 10
Hence, {6, 8, 10} is a triplet.
(c) {6, 8, 15}
Here, ![(6)^{2} + (8)^{2} = 36 + 64 = 100\\\implies H = \sqrt{100} = 10](https://tex.z-dn.net/?f=%286%29%5E%7B2%7D%20%20%2B%20%288%29%5E%7B2%7D%20%20%20%3D%2036%20%2B%2064%20%20%3D%20100%5C%5C%5Cimplies%20H%20%3D%20%5Csqrt%7B100%7D%20%20%3D%20%2010)
So, third side of the triangle 10 ≠ 15
Hence, {6, 8, 15} is NOT a triplet.
(d) {5, 7, 13}
Here, ![(5)^{2} + (7)^{2} = 25 + 49 = 74\\\implies H = \sqrt{74} = 8.60](https://tex.z-dn.net/?f=%285%29%5E%7B2%7D%20%20%2B%20%287%29%5E%7B2%7D%20%20%20%3D%2025%20%2B%2049%20%20%3D%2074%5C%5C%5Cimplies%20H%20%3D%20%5Csqrt%7B74%7D%20%20%3D%20%208.60)
So, third side of the triangle 8.60 ≠ 13
Hence, {5, 7, 13} is NOT a triplet.