Question

Apply the triangle inequality theorem to determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters and 9 millimeters

Answer 1

The triangle inequality theorem states that for three sides \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be satisfied:

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

Let's apply these conditions to the lengths 7 mm, 8 mm, and 9 mm.

Let:
- \( a = 7 \) mm
- \( b = 8 \) mm
- \( c = 9 \) mm

Now, we can check each of the inequalities:

1. \( a + b > c \):
\[
7 + 8 > 9 \quad \Rightarrow \quad 15 > 9 \quad \text{(True)}
\]

2. \( a + c > b \):
\[
7 + 9 > 8 \quad \Rightarrow \quad 16 > 8 \quad \text{(True)}
\]

3. \( b + c > a \):
\[
8 + 9 > 7 \quad \Rightarrow \quad 17 > 7 \quad \text{(True)}
\]

Since all three conditions are satisfied, it is possible to form a triangle with sides of lengths 7 mm, 8 mm, and 9 mm.