Final answer:
The sides with lengths of 6, 4, and 2 cannot form a triangle because they do not satisfy the Triangle Inequality Theorem, which requires the sum of the lengths of any two sides to be greater than the length of the third side.
Step-by-step explanation:
The sides of a triangle must satisfy a specific condition known as the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side.
Therefore, to determine if sides measuring 6, 4, and 2 can form a triangle, we perform the following checks
6 + 4 > 2
6 + 2 > 4
4 + 2 > 6
While the first two checks are true, the third check, 4 + 2 > 6, is false. This means that the sides 6, 4, and 2 cannot form a triangle, because the sum of the lengths of the two smaller sides is not greater than the length of the longest side.
When considering the possibility of these lengths forming a right triangle, we can refer to the Pythagorean theorem, which states that for a right triangle with sides of lengths a, b, and hypotenuse c, the following must be true:
a² + b² = c². However, in this case, 2² + 4² is not equal to 6², so even as a right triangle, these side lengths are not possible.