Final answer:
For the sides of a triangle in arithmetic progression, the smallest integer ratio adhering to the triangle inequality theorem is 1:2:3, with 'a' as the smallest side, 'a+d' as the middle side, and 'a+2d' as the largest side.
Step-by-step explanation:
The question pertains to the properties of a triangle whose sides are in arithmetic progression (AP). If the sides of a triangle are in AP, it means there is a common difference 'd' such that each side is 'd' units longer than the previous one. Let's denote the lengths of the sides as 'a', 'a+d', and 'a+2d'. To ensure that these sides form a triangle, they must satisfy the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. According to this, the following inequalities must hold true: a + (a + d) > (a + 2d), (a + d) + (a + 2d) > a, and a + (a + 2d) > (a + d).
By simplifying these inequalities, we find that the common difference 'd' must be less than 'a' and greater than 0. Thus, the lowest integer ratio of the lengths of the triangle's sides that satisfies these conditions is 1:2:3, which is option A.