SOLUTION: Prove or disprove: any two right triangles with the same length hypotenuse have the same area.

We will disprove it by giving one counter-example.

First of all I know that a right triangle can have
legs 3 and 4, with hypotenuse 5.  I also know that 
a right triangle can have legs 5 and 12, with a
hypotenuse of 13.  I also know that if I multiply
all three sides' lengths by a constant, I will get a
triangle similar to the original one. So I reason
that if I multiply the sides of a 3,4,5 right triangle
by 13, and the sides of a 5,12,13 triangle by 5, I
will get two right triangles with the same hypotenuse
of 5*13 or 65.  


So we consider two triangles.  

Let the first triangle have sides 39, 52, and longest side 65.
Let the second triangle have sides 25, 60, and longest side 65.



First, we must prove that these are both right triangles
We will use the inverse of the Pythagorean theorem.

We prove the first one is a right triangle by showing that
the Pythagorean equation applies:


if and only if

if and only if

which is true.

We also prove the second one is a right triangle
the same way:


if and only if

if and only if

which is true.

Next we calculate the area of the first one:






Finally we calculate the area of the second one:






The areas are not equal.  So we have disproved that
any two right triangles with the same hypotenuse have 
the same area, for here are two right triangles with
the same length hypotenuse, 65, yet they do not have the
same area.

Edwin