To create the flower gardens, Wendell bought six pieces of wood. Pieces A and B are 6 feet long, pieces C and D are 8 feet long, piece E is 3 feet long, and piece F is 2 feet long.
Part A
Can Wendell make a triangular garden using pieces A, B, and F? Why or why not?



Part B
Can Wendell make a triangular garden using pieces D, E, and F? Why or why not?



Part C
Can Wendell make a rectangular garden with the pieces of wood he has? If yes, which pieces can he use?



Part D
Describe how Wendell can use all six pieces of wood to create either two rectangular gardens or two triangular gardens. Assume the gardens do not share a common side.



Part E
Wendell’s dog, Jordan, was getting in his way as he worked in the backyard. So, Wendell chained him to a pole. If the chain is 12 feet long, about how much area does Jordan have to walk around?

Let's go through each part step by step.

### Part A
**Question: Can Wendell make a triangular garden using pieces A, B, and F? Why or why not?**

To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. We can use the triangle inequality theorem to check:

- Pieces A and B are 6 feet each, and piece F is 2 feet.
  
Let's check the inequalities:
1. \(6 + 6 > 2\) (True)
2. \(6 + 2 > 6\) (True)
3. \(6 + 2 > 6\) (True)

Since all inequalities are satisfied, **Wendell can make a triangular garden using pieces A, B, and F**.

### Part B
**Question: Can Wendell make a triangular garden using pieces D, E, and F? Why or why not?**

Using the triangle inequality theorem:

- Pieces D is 8 feet, E is 3 feet, and F is 2 feet.

Check the inequalities:
1. \(8 + 3 > 2\) (True)
2. \(8 + 2 > 3\) (True)
3. \(3 + 2 > 8\) (False)

Since \(3 + 2\) is not greater than \(8\), **Wendell cannot make a triangular garden using pieces D, E, and F**.

### Part C
**Question: Can Wendell make a rectangular garden with the pieces of wood he has? If yes, which pieces can he use?**

To make a rectangular garden, opposite sides need to be of equal length.

- Pieces available: 6 feet (A), 6 feet (B), 8 feet (C), 8 feet (D), 3 feet (E), 2 feet (F).

Wendell can make a rectangular garden using pairs of equal-length pieces:
1. Use pieces A (6 feet) and B (6 feet) for the shorter sides.
2. Use pieces C (8 feet) and D (8 feet) for the longer sides.

Thus, **Wendell can make a rectangular garden using pieces A, B, C, and D**.

### Part D
**Question: Describe how Wendell can use all six pieces of wood to create either two rectangular gardens or two triangular gardens. Assume the gardens do not share a common side.**

**Option 1: Two rectangular gardens**

1. First rectangular garden:
   - Pieces A (6 feet) and B (6 feet) for the shorter sides.
   - Pieces C (8 feet) and D (8 feet) for the longer sides.
   
2. Second rectangular garden:
   - This is not possible with the remaining pieces E (3 feet) and F (2 feet).

**Option 2: Two triangular gardens**

1. First triangular garden:
   - Pieces A (6 feet), B (6 feet), and F (2 feet) (as checked in Part A).

2. Second triangular garden:
   - Pieces C (8 feet), D (8 feet), and E (3 feet).
   
Check the inequalities for the second triangle:
1. \(8 + 8 > 3\) (True)
2. \(8 + 3 > 8\) (True)
3. \(8 + 3 > 8\) (True)

Since all inequalities are satisfied, the second triangular garden is possible.

Thus, **Wendell can create two triangular gardens using pieces A, B, F for the first, and C, D, E for the second**.

### Part E
**Question: If the chain is 12 feet long, about how much area does Jordan have to walk around?**

Jordan, being chained to a pole with a 12-foot chain, can walk around in a circle with a radius of 12 feet. The area of a circle is given by \( \pi r^2 \).

So, the area \( A \) is:
\[ A = \pi \times (12)^2 \approx 3.14 \times 144 \approx 452.16 \]

Thus, **Jordan has about 452.16 square feet to walk around**.

Do you want details or have any questions about these explanations?

### Suggested Questions:
1. How do you find the area of more complex shapes?
2. Can you explain more about the triangle inequality theorem?
3. How would you approach a problem involving more than three sides?
4. What other shapes can you make with different lengths of pieces?
5. How can you use different methods to calculate area and perimeter?

### Tip:
When checking the feasibility of forming a shape, always ensure to use geometric properties and theorems, like the triangle inequality theorem, to verify the possibility of the structure.

Explore how to solve geometry problems involving triangles, rectangles, and circles with detailed step-by-step solutions. Covering key concepts such as the Triangle Inequality Theorem and the calculation of area for circles. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation