Question

What number of triangles can be formed using the set of three side lengths 11in,15in,6in

One unquie triangle
Many triangles
No triangles

Answer 1

To determine whether a triangle can be formed with the side lengths of 11 inches, 15 inches, and 6 inches, we can apply the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following conditions must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Let's analyze the given side lengths: \( a = 6 \), \( b = 11 \), and \( c = 15 \).

1. Check \( a + b > c \):
\[
6 + 11 = 17 > 15 \quad \text{(True)}
\]

2. Check \( a + c > b \):
\[
6 + 15 = 21 > 11 \quad \text{(True)}
\]

3. Check \( b + c > a \):
\[
11 + 15 = 26 > 6 \quad \text{(True)}
\]

All three conditions of the triangle inequality are satisfied. Therefore, a triangle can indeed be formed with the side lengths of 11 inches, 15 inches, and 6 inches.

Since the side lengths are fixed, only **one unique triangle** can be formed with these dimensions.

So, the answer is: **One unique triangle**.