The Pythagorean Theorem - Free Printable

Problem Analysis:

The image shows two sets of geometric pieces labeled as "Puzzle Proof One" and "Puzzle Proof Two." The task appears to involve arranging these pieces to form specific shapes or prove a mathematical concept, likely related to the Pythagorean theorem. Let's break this down step by step.

#### Puzzle Proof One:
- This section contains five labeled pieces: 1, 2, 3, 4, and 5.
- Piece 1 is a large square.
- Pieces 2, 3, 4, and 5 are right triangles and smaller squares.

#### Puzzle Proof Two:
- This section contains seven labeled pieces: 6, 7, 8, 9, 10, 11, and 12.
- Pieces 6, 7, 8, 9, 10, and 11 are right triangles and smaller squares.
- Piece 12 is a large square.

Objective:

The goal is to arrange the pieces from each set to demonstrate the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of a right triangle, and \(c\) is the hypotenuse.

Solution Approach:

#### Step 1: Understanding Puzzle Proof One
1. Identify the Shapes:
- Piece 1 is a large square with side length \(c\), representing \(c^2\).
- Pieces 2, 3, 4, and 5 are four right triangles and two smaller squares. The triangles have legs \(a\) and \(b\), and the smaller squares have areas \(a^2\) and \(b^2\).

2. Arrange the Pieces:
- Place the four right triangles (pieces 2, 3, 4, and 5) around the central square (piece 1) such that they fit perfectly into the larger square.
- The remaining space inside the larger square will be occupied by two smaller squares, one with area \(a^2\) and the other with area \(b^2\).

3. Verification:
- The total area of the large square (piece 1) is \(c^2\).
- The total area of the four triangles is \(2 \times \left(\frac{1}{2}ab\right) = ab\).
- The remaining area is the sum of the two smaller squares, which is \(a^2 + b^2\).
- Therefore, the arrangement visually demonstrates:
\[
a^2 + b^2 = c^2
\]

#### Step 2: Understanding Puzzle Proof Two
1. Identify the Shapes:
- Piece 12 is a large square with side length \(c\), representing \(c^2\).
- Pieces 6, 7, 8, 9, 10, and 11 are six right triangles and two smaller squares. The triangles have legs \(a\) and \(b\), and the smaller squares have areas \(a^2\) and \(b^2\).

2. Arrange the Pieces:
- Place the six right triangles (pieces 6, 7, 8, 9, 10, and 11) around the central square (piece 12) such that they fit perfectly into the larger square.
- The remaining space inside the larger square will be occupied by two smaller squares, one with area \(a^2\) and the other with area \(b^2\).

3. Verification:
- The total area of the large square (piece 12) is \(c^2\).
- The total area of the six triangles is \(3 \times \left(\frac{1}{2}ab\right) = \frac{3}{2}ab\).
- The remaining area is the sum of the two smaller squares, which is \(a^2 + b^2\).
- Therefore, the arrangement visually demonstrates:
\[
a^2 + b^2 = c^2
\]

Final Answer:

Both sets of pieces ("Puzzle Proof One" and "Puzzle Proof Two") can be arranged to visually demonstrate the Pythagorean theorem:
\[
\boxed{a^2 + b^2 = c^2}
\]