The combined area of the shaded triangles in figure one is 2ab. We have four congruent right triangles given, with legs a and b. The formula to calculate the area of a right triangle is (leg * leg) / 2. So the area of one of the triangles is ab/2. We have four of these triangles, so ab/2 * 4 = 2ab.
The area of the unshaded square in figure one can be represented as c^2. Since the side of the square is given as c, its area must be c^2 (calculate the area of a square by multiplying its side length by itself).
The combined area of the unshaded squares in figure 2 can be represented by a^2 + b^2. Again, we find the areas of each square by multiplying the side lengths togther: (a * a) + (b * b) = a^2 + b^2.
Finally the areas of the squares in figure one and figure two show that a^2 + b^2 = c^2. This is because the shaded triangles in figure one take up the same area as that of figure two. And since the total figure areas are the same, the unshaded parts of each figure must be equal to each other. Therefore, we have a^2 + b^2 = c^2. This is also known as the Pythagorean Theorem.