The task involves demonstrating the
Pythagorean Theorem using a
proof by dissection. This method visually shows how the areas of the squares on the legs of a right triangle can be rearranged to fit exactly into the square on the hypotenuse.
Problem Overview:
The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the lengths of the legs of a right triangle, and \(c\) is the length of the hypotenuse.
In this problem, you are given geometric pieces (triangles and squares) that need to be rearranged to fill:
1. Two smaller squares (corresponding to \(a^2\) and \(b^2\)).
2. A larger square (corresponding to \(c^2\)).
Solution Explanation:
#### Step 1: Understand the Pieces
The image contains:
- Two blue triangles.
- One red triangle.
- One green triangle.
- Two smaller white squares (one in each of the two smaller squares).
- One large white square (in the larger square).
These pieces are designed to fit perfectly into both configurations:
1. Filling the two smaller squares.
2. Filling the larger square.
#### Step 2: Fill the Two Smaller Squares
- The two blue triangles and one of the smaller white squares will fit into one of the smaller squares.
- The red triangle and the other smaller white square will fit into the second smaller square.
This arrangement visually represents the areas \(a^2\) and \(b^2\).
#### Step 3: Fill the Larger Square
- All the pieces (two blue triangles, one red triangle, one green triangle, and the two smaller white squares) can be rearranged to fit perfectly into the larger square.
This arrangement visually represents the area \(c^2\).
#### Step 4: Verify the Areas
- The total area of the two smaller squares is the sum of the areas of the triangles and the smaller white squares.
- When rearranged, these pieces exactly fill the larger square, showing that:
\[
a^2 + b^2 = c^2
\]
Final Answer:
By dragging and rotating the pieces as instructed, you can see that the areas of the two smaller squares (corresponding to \(a^2\) and \(b^2\)) can be reassembled to fill the larger square (corresponding to \(c^2\)). This visually proves the Pythagorean Theorem.
\[
\boxed{a^2 + b^2 = c^2}
\]
Parent Tip: Review the logic above to help your child master the concept of pythagorean puzzle worksheet.
