Final answer:
To solve for the variable x and the lengths of the sides of triangle ARS, we use the expressions given for the sides and check against the triangle inequality theorem. By substituting the potential values of x, we conclude that option A provides the correct dimensions for a valid triangle.
Step-by-step explanation:
In the context of the given problem, the sides of the triangle ARS are defined in terms of the variable x. To find the value of x, we need to recall that the sum of all sides in a triangle equals the perimeter. Therefore, we add the expressions for sides RS, ST, and RT. The equation will look like this:
RS + ST + RT = Perimeter of triangle ARS
(9x – 13) + (10x – 3) + (4x + 2) = Total length. After simplifying, we have:
23x – 14 = Total length
Since we do not have the total length of the triangle ARS, we can't determine the exact value of x from the given options. However, we can check which option is correct by substituting the values of x in place and seeing if the sides satisfy the triangle inequality theorem.
Option A, for instance, gives us x = 5, resulting in RS = 32, ST = 47, and RT = 22. We add these values to check if they satisfy the triangle inequality theorem:
32 + 47 > 22
47 + 22 > 32
22 + 32 > 47
All these inequalities hold true, thus option A is the correct answer and the triangle can exist with these side lengths.