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Rationalize This

Problem Statement
A rational right triangle is a triangle where all the sides are rational numbers and one of the angles is a right angle.

Find a rational right triangle such that the length of the hypotenuse is numerically equal to the area of the triangle, and the perimeter of the triangle is a prime integer.

Problem setup

This problem asks to find the measures of a triangle in which the length of the hypotenuse is the same measure as the area of the triangle. The perimeter of the triangle must be a prime number and the lengths of the sides must be rational numbers.


Plans to Solve/Investigate the Problem

I am going to open GSP and construct a right triangle. I will then measure the lengths of the sides and the area and perimeter of that triangle. I’ll then experiment to see what happens when I change these measures.

Investigation/Exploration of the Problem

When I opened GSP I constructed a right triangle with the following measures:


I then began dragging the points around and looking at the measures of the sides and the area.







 By experimentation, I found the following triangle.:

As I was dragging the points to create this triangle, I observed several things.

1.     The size of the lengths of each side cannot be equal for this to work.

2.     As the size of the lengths increase, the area increases, so the triangles must have a small area in order to fit the requirements of the problem.

3.     The length of EF is approximately one tenth of the length of DF. This was quite interesting to me. I found that because of the formula bh/2, this would make sense that the base would be one tenth of the height.