Polygon - Free Printable

Problem Description:

The image shows two geometric figures:
1. A regular pentagon on the left.
2. A pentagon inscribed in a square on the right, with one of its vertices marked in red.

The task is to determine the relationship between the pentagon and the square, specifically how the pentagon is positioned within the square.

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Solution:

#### Step 1: Analyze the Regular Pentagon
- A regular pentagon has five equal sides and five equal interior angles. Each interior angle of a regular pentagon is given by:
$$
\text{Interior angle} = \frac{(5-2) \times 180^\circ}{5} = 108^\circ.
$$

#### Step 2: Inscribed Pentagon in a Square
- The pentagon on the right is inscribed in a square. This means all vertices of the pentagon lie on the boundary of the square.
- The pentagon is not necessarily regular, as its sides and angles may vary. However, it is constrained by the square's edges.

#### Step 3: Identify Key Features
- The pentagon has one vertex marked in red. This vertex appears to be at the intersection of two edges of the square.
- The pentagon's other vertices are distributed along the edges of the square, but not necessarily symmetrically.

#### Step 4: Relationship Between the Pentagon and the Square
- The pentagon is inscribed in the square, meaning it fits entirely within the square, and its vertices touch the square's edges.
- The pentagon's shape is determined by how its vertices are positioned along the square's edges. Since the pentagon is not regular, its sides and angles can vary, but they must still satisfy the condition that all vertices lie on the square's perimeter.

#### Step 5: Conclusion
- The pentagon is a specific configuration where its vertices are placed along the edges of the square. The exact shape of the pentagon depends on the positions of its vertices, but it is always constrained by the square's boundaries.

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Final Answer:

The pentagon is inscribed in the square, meaning all its vertices lie on the edges of the square. The specific configuration of the pentagon depends on the placement of its vertices along the square's edges.

$$
\boxed{\text{The pentagon is inscribed in the square.}}
$$