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 Therom explain why or why not.

Answer 1

To determine whether three objects can form a triangle using the Triangle Inequality Theorem, we first need to have three objects with measurable lengths. Let's say we have the following three objects:

1. A stick that is 5 inches long.
2. A ribbon that is 7 inches long.
3. A book that is 3 inches thick.

Next, we apply the Triangle Inequality Theorem, which states that for any three lengths \(a\), \(b\), and \(c\), the following conditions must be met for them to form a triangle:

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

In this case, let’s assign:
- \(a = 5\) inches (stick)
- \(b = 7\) inches (ribbon)
- \(c = 3\) inches (book)

Now we check the inequalities:

1. \(a + b > c \quad \Rightarrow \quad 5 + 7 > 3 \quad \Rightarrow \quad 12 > 3 \) (True)
2. \(a + c > b \quad \Rightarrow \quad 5 + 3 > 7 \quad \Rightarrow \quad 8 > 7 \) (True)
3. \(b + c > a \quad \Rightarrow \quad 7 + 3 > 5 \quad \Rightarrow \quad 10 > 5 \) (True)

Since all three conditions are satisfied, the objects can indeed form a triangle. Thus, using the stick, ribbon, and book, we can create a triangle.