# Pythagorean Theorem $a^2 + b^2 = c^2$ ![[Pasted image 20240627224803.png]] ## Proof Here we have a right triangle with sides $\Delta abc$ with an area of $\frac{ab}{2}$. ![[mikeyMannMade_triangle.png]] Lets make this square with said triangle. ![[mikeyMannMade_trianglex4.png]] We now have this large square with an area of $(a+b)^2$. In standard form the area looks like $a^2 + 2ab +b^2$. ![[mikeyMannMade_triangleLarge.png]] And this small square with area $c^2$. ![[mikeyMannMade_triangleSmall.png]] With this in mind we can state that the area of the large square is equivalent to the area of the small square plus the area of the 4 triangles. Therefore: $a^2+2ab+b^2=c^2+4(\frac{ab}{2})$ $a^2+2ab+b^2=c^2+\cancel4(\frac{ab}{\cancel2})$ $a^2+2ab+b^2=c^2+2(ab)$ $a^2+\cancel{2ab}+b^2=c^2+\cancel{2ab}$ $\boxed{a^2+b^2=c^2}$