Drag a model next to the matching description to complete this proof of the Pythagorean theorem. Show that a square with area c² can be transformed into two squares with areas a² and b².
Each side length of a square is c so it has an area of c².
One small square and 4 congruent right triangles make a square.
Moving two of the triangles changes the figure’s shape, but not its area.
This figure is made up of one square with an area of a² and one square with an area of b².
a² + b² = c².
To prove the Pythagorean theorem ( a^2 + b^2 = c^2 ) using your description, we can follow these steps:
1. Start with a large square: This square has a side length of ( c ), thus its area is ( c^2 ).
2. Divide the square: Inside this square, you can place one small square with an area of ( a^2 ) and four congruent right triangles.
3. Rearranging: When you move two of the triangles from the original configuration, you can arrange them along one side of the smaller square of area ( a^2 ), effectively creating the second square with an area ( b^2 ).
4. Area consistency: The total area of the rearranged figure – the square with area ( a^2 ) and the square with area ( b^2 ) plus the triangles – remains unchanged at ( c^2 ).
Thus, you can visually see that ( a^2 + b^2 = c^2 ) holds true because you’ve reconfigured the area without changing it.
This demonstration visually and mathematically supports the Pythagorean theorem, showing how the areas relate to each other when you manipulate the shapes. For a more detailed visual representation or deeper insight, consider checking out extended services for further assistance.