Question

Find three objects (sticks, utensils, yarn, toys, books, etc.) that could represent
the sides of a triangle. Can you make a triangle with these three objects? Use the
Triangle Inequality Theorem explain why or why not
measurements of each object:
Application of Triangle Inequality Theorem:

Answer 1

To explore whether three objects can represent the sides of a triangle using the Triangle Inequality Theorem, we can first choose three objects and measure their lengths. Let's consider the following objects:

1. **Stick A**: 5 cm
2. **Stick B**: 7 cm
3. **Stick C**: 10 cm

### Application of the Triangle Inequality Theorem
The Triangle Inequality Theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold true:

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

In our case, we assign:
- \( a = 5 \) cm (Stick A)
- \( b = 7 \) cm (Stick B)
- \( c = 10 \) cm (Stick C)

Let's check the Triangle Inequality:

1. **Check \( a + b > c \)**:
\[
5 + 7 > 10 \\
12 > 10 \quad \text{(True)}
\]

2. **Check \( a + c > b \)**:
\[
5 + 10 > 7 \\
15 > 7 \quad \text{(True)}
\]

3. **Check \( b + c > a \)**:
\[
7 + 10 > 5 \\
17 > 5 \quad \text{(True)}
\]

### Conclusion
All three inequalities are satisfied, which indicates that Stick A (5 cm), Stick B (7 cm), and Stick C (10 cm) can indeed form a triangle.

If we were to select different lengths or if one of the inequalities failed, forming a triangle would not be possible. In this case, we have successfully demonstrated that the selected objects can represent the sides of a triangle according to the Triangle Inequality Theorem.