Three square tiles are laid out on a table so that they make a solid pattern.
Each tile must touch at least one other tile along an entire edge. The
squares all have sides of length 1.
(a) Draw all possible patterns.
(b) What are the possible perimeters of the figures that can be formed?
(c) Which figure has the least perimeter?

Let's break down the problem step by step:

### (a) Drawing all possible patterns:
We are given three square tiles, each with side length 1, and they must touch along an entire edge. There are several possible ways to arrange these tiles while keeping that condition in mind. Here are the patterns:

1. **All tiles in a straight line**: This forms a 1x3 rectangle.
   
   ```
   [ ] [ ] [ ]
   ```

2. **Two tiles in a straight line, one attached to the side**: This forms an L-shape.
   
   ```
   [ ] [ ]
   [ ]
   ```

3. **Three tiles forming a "T"-shape**: The middle tile has one tile attached to each of its sides.

   ```
      [ ]
   [ ][ ]
   ```

These are the main possible configurations where each square touches another along an entire edge.

### (b) What are the possible perimeters of the figures?

For each of the three patterns, let's calculate the perimeter:

1. **Straight line (1x3 rectangle)**:
   - This figure has a length of 3 and a width of 1.
   - Perimeter = \( 2 \times (\text{length} + \text{width}) = 2 \times (3 + 1) = 8 \).

2. **L-shape**:
   - This figure has a length of 2 and a width of 2.
   - Perimeter = \( 2 \times (\text{length} + \text{width}) = 2 \times (2 + 2) = 8 \).

3. **T-shape**:
   - This figure also has a length of 2 and a width of 2.
   - Perimeter = \( 2 \times (\text{length} + \text{width}) = 2 \times (2 + 2) = 8 \).

Thus, the possible perimeter of each pattern is **8 units**.

### (c) Which figure has the least perimeter?

From the previous part, we see that all the figures have the same perimeter, which is 8 units. Therefore, **all figures have the least perimeter of 8**.

---

Would you like any further clarifications, or do you want a more detailed breakdown of the problem?

Here are five related questions to deepen your understanding:

1. What would the perimeter be if four tiles were arranged in a similar way?
2. How would the perimeter change if tiles of different lengths were used?
3. Can you think of a different arrangement that still satisfies the edge-touching condition?
4. What if the tiles were not squares but rectangles? How would this affect the possible perimeters?
5. How does the perimeter of a shape relate to its area for different configurations?

**Tip**: When calculating the perimeter of composite shapes, focus on counting only the edges that form the outer boundary of the figure!

Three square tiles are laid out on a table so that they make a solid pattern. Each tile must touch at least one other tile along an entire edge. The squares all have sides of length 1. (a) Draw all possible patterns. (b) What are the possible perimeters of the figures that can be formed? (c) Which figure has the least perimeter?

Explore the possible patterns formed by arranging three square tiles with side lengths of 1. Learn how to calculate the perimeters of various composite shapes such as a straight line, L-shape, and T-shape, all having a perimeter of 8 units. Suitable for students in Grades 5-7.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation