To determine if a unique triangle can be drawn with given side lengths, you need to verify if the side lengths satisfy the Triangle Inequality Theorem. This theorem states that for any three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Let's apply this theorem to the given side lengths:
Given side lengths:
- [tex]\( a = 8 \)[/tex] inches
- [tex]\( b = 12 \)[/tex] inches
- [tex]\( c = 25 \)[/tex] inches
According to the Triangle Inequality Theorem, the following three conditions must all be true for the sides to form a triangle:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Let's check each condition:
1. Check if [tex]\( a + b > c \)[/tex]:
[tex]\[
8 + 12 > 25 \implies 20 > 25 \quad \text{(False)}
\][/tex]
2. Check if [tex]\( a + c > b \)[/tex]:
[tex]\[
8 + 25 > 12 \implies 33 > 12 \quad \text{(True)}
\][/tex]
3. Check if [tex]\( b + c > a \)[/tex]:
[tex]\[
12 + 25 > 8 \implies 37 > 8 \quad \text{(True)}
\][/tex]
From the above conditions, we see that the first condition ([tex]\( a + b > c \)[/tex]) is not satisfied because 20 is not greater than 25. If any one of these conditions is not satisfied, the given side lengths cannot form a triangle.
Since the first condition fails, it is impossible to form a triangle with side lengths 8 inches, 12 inches, and 25 inches.
Hence, the number of unique triangles that can be drawn with these side lengths is:
[tex]\[
\boxed{0}
\][/tex]
