#1**+1 **

By the Pythagorean Theorem, we know that x^{2} + h^{2} = a^{2} and y^{2} + h^{2} = b^{2}

which, if we add them together, we get: x^{2} + h^{2} + y^{2} + h^{2} = a^{2} + b^{2}

However, we want to show that (x + h)^{2} + (y + h)^{2} = (a + b)^{2}

and, this is: ( x^{2} + 2xh + h^{2} ) + ( y^{2} + 2yh + h^{2} ) = a^{2} + 2ab + b^{2}

So, somehow we will need to show tha 2xh + 2yh = 2ab

We can do this by finding the area of the triangle in two different ways:

Area = ½ab and the area is also Area = ½(x +y)h

So: ½(x +y)h = ½ab

---> (x + y)h = ab

---> xh + yh = ab

---> 2xh + 2yh = 2ab

Adding this to: x^{2} + h^{2} + y^{2} + h^{2} = a^{2} + b^{2}

we get: x^{2} + 2xh + h^{2} + y^{2} + 2yh + h^{2} = a^{2} + 2ab + b^{2}

which is: (x + h)^{2} + (y + h)^{2} = (a + b)^{2}

geno3141 Jun 5, 2020