A triangle has a perimeter of 13 units. The two shorter sides are consecutive integers with lengths x and x + 1. Based on triangle inequality conditions, determine which of the following could be the length of the third side.

Introduction / Context:
This question tests understanding of triangle inequality rules. For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Given the perimeter and two sides in terms of variables, the possible value of the remaining side can be deduced logically.

Given Data / Assumptions:

• Total perimeter = 13
• Two sides are x and x + 1
• Third side = 13 - (x + x + 1) = 12 - 2x

Concept / Approach:
Use triangle inequality: the sum of the two shorter sides must be greater than the third side. Check integer values of x that satisfy all triangle inequalities and see which option matches the possible third side length.

Step-by-Step Solution:
Let sides be x, x + 1, and 12 - 2x For a valid triangle: x + (x + 1) > (12 - 2x) 2x + 1 > 12 - 2x 4x > 11 => x > 2.75 Try x = 3: sides are 3, 4, and 6 (valid triangle) Thus third side = 6 But 6 is not shorter than x and x+1, violating the given condition Try x = 4: sides are 4, 5, and 4 (valid)

Verification / Alternative check:
For x = 4, all triangle inequalities are satisfied and the perimeter is 13. The third side equals 4, which is consistent with the problem statement.

Why Other Options Are Wrong:
6 and 8: violate triangle inequality or ordering of sides. None of the above: incorrect because 4 is valid. 5: does not match the computed third side.

Common Pitfalls:
Ignoring the triangle inequality conditions. Forgetting that x and x+1 are the shorter sides. Arithmetic errors while forming the third side.

Final Answer:
The possible length of the third side is 4