An isosceles triangle has at least two congruent sides. The perimeter of a certian isosceles triangle is at most 12 in. The length of each of the two congruent sides is 5 in. What are the possible lengths of the remaning side?
Let $s$ be the length of the remaining side. If we attempt to set up the perimeter, we get $5+5+s<12 \Rightarrow 10+s<12 \Rightarrow s<2,$ so our remaining side must be less than $2$ inches in length. However, if we set our remaining side to have length $0,1,$ or $2$ inches, the triangle would degenerate into a line segment and not exist at all, so our possible answer is $\boxed{}$ None.
Since the isosceles triangle has two congruent sides of length 5 in, the remaining side must have a length less than or equal to the sum of these two sides. Let's denote the length of the remaining side as "x".
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
So, we have the following inequality:
5 + 5 > x
Simplifying, we get:
10 > x
Since the perimeter of the triangle is at most 12 in, we can also write:
2(5) + x ≤ 12
10 + x ≤ 12
Subtracting 10 from both sides, we have:
x ≤ 2
Therefore, the possible lengths of the remaining side of the isosceles triangle are any value that is less than or equal to 2 inches.
To find the possible lengths of the remaining side of the isosceles triangle, we need to consider the relationship between the lengths of the sides in an isosceles triangle.
In an isosceles triangle, the two congruent sides are equal in length. Let's call the length of each congruent side "x" and the length of the remaining side "y".
According to the given information, the length of each congruent side is 5 inches. Therefore, x = 5 inches.
The perimeter of a triangle is the sum of the lengths of its sides. In this case, the perimeter of the triangle is at most 12 inches.
Perimeter = x + x + y
Perimeter = 5 + 5 + y
Perimeter = 10 + y
The perimeter is at most 12 inches, so we have the inequality:
10 + y ≤ 12
To find the possible lengths of the remaining side, we can solve this inequality:
y ≤ 12 - 10
y ≤ 2
Therefore, the possible lengths of the remaining side (y) are less than or equal to 2 inches. So, the remaining side can be any length less than or equal to 2 inches.