Title
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.
.
