An isosceles triangle has at least two congruent sides. The perimeter of a certain isosceles triangle is at most 12 inches. The length of the two congruent sides is 5 inches. What is the possible length of the remaining side? A. S is greater than or equal to 5 but is less than 12 B. S is less than or equal to 12 but greater than 0 C. S is less than or equal to 10 but greater than 0 D. S is less than or equal to 2 but greater than 0

The perimeter of a triangle is the sum of the lengths of its three sides. Given that we have an isosceles triangle with two congruent sides each measuring 5 inches, the perimeter P for the triangle can be expressed as:

P = Side1 + Side2 + Base P = 5 + 5 + Base P = 10 + Base

Since the perimeter is at most 12 inches, we have:

10 + Base ≤ 12

Now, to find the possible length of the base, we must solve for Base. Subtracting 10 from both sides of the inequality gives us:

Base ≤ 12 - 10 Base ≤ 2

Therefore, the base must be less than or equal to 2 inches for the perimeter to be at most 12 inches.

However, since we're dealing with the length of a side of a triangle, it must be greater than 0. Also, we need to keep in mind 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.

In our case, because the triangle is isosceles and the two congruent sides are each 5 inches long, the base must also be less than the combined length of these two sides (because if it were equal or greater, it would not form a triangle).

This means we have an additional requirement:

Base < 5 + 5 Base < 10

Putting both conditions together (Base ≤ 2 and Base < 10), the possible length of the base must adhere to the stricter constraint, which is:

Base ≤ 2

This means that the length of the remaining side must be greater than 0 but less than or equal to 2 inches.

The correct answer reflecting these constraints is: D. S is less than or equal to 2 but greater than 0