Short Answer 
The problem involves analyzing areas of shapes in two figures, leading to the conclusion that the equality of shaded triangle areas establishes the relationship ( a² + b² = c² ). This reinforces the Pythagorean Theorem, which connects the dimensions of right triangle sides with the respective square areas.
Step 1: Understand the Area Representation
In the problem, we are given figures that illustrate areas represented by squares and triangles. Each triangle’s area in Figure 1 corresponds to shaded triangles in Figure 2, indicating a relationship between these two figures. Additionally, the unshaded squares are represented in terms of their side lengths:
- Figure 1 has an unshaded square area represented by c².
- Figure 2 includes two unshaded squares with areas a² and b².
Step 2: Analyze the Equality of Areas
The primary relationship established is that the shaded triangles’ areas in both figures are equal. This leads us to derive an important equation relating the areas of the shapes. The unshaded areas help us connect these two figures through the formula:
- The area of the unshaded square in Figure 1 is c².
- The combined area of unshaded squares in Figure 2 equals a² + b².
Step 3: Conclude the Relationship
With the equality of the shaded triangle areas established and the areas of the squares represented accurately, we can conclude that the equation holds true: a² + b² = c². This establishes a fundamental connection between the dimensions of the triangles and the corresponding square areas, reinforcing the geometric principles at play:
- This equation represents the Pythagorean Theorem.
- It shows the relationship between the sides of right triangles.