# Right angled triangles with integer areas.

I've recently been working on the following problem:

Is there a right angled triangle with rational side lengths and an area of $1$?

I was told I wouldn't be able to solve it, but this was simply an exercise to see how I go about solving problems.

I indeed was not able to solve, but was able to reduce it to a problem with an elliptic curve that I suspect relates to Fermat's last Theorem. However, I suspect there is a more subtle and simpler approach to the problem. So what exactly is the answer?

I will give a hint.Let one side of the triangle be a/b.As i had mentioned above to get the area as one the length of 2nd side must be 2*b/a.We get the hypotenuse as sqrt(4b^4+a^4)/ab.Since we take the a,b values as integers we get the denominator a integer value.now solve the numerator to get an integer value which give you yhe required answer

Let the right triangle have sides of length $a$, $b$, and $c$ (where $c$ is the length of the hypotenuse) and suppose $a$, $b$, and $c$ are all rational numbers.

By previous knowledge we have the equation $(a^2 – b^2)^2 = (a^2 + b^2)^2 – 4a^2b^2$.

We also know the area of the triangle is $1$, so therefore $\frac {ab}{2} = 1$. Multiplying both sides by $2$ gives $ab = 2$.

Using the Pythagorean theorem we have $(a^2 – b^2)^2 = c^4 – 16$. However, by previous knowledge we know a theorem of Fermat which states that the difference of two rational numbers raised to a power of $4$ cannot be a square of a rational number. Therefore, we have a contradiction and the proof is complete.

Special thanks to @tatan @lulu @GerryMyerson for helping in the proof.

• Your proof is not correct....side lengths are rational....not integer...so you cannot assume $x=2,y=1$...there are infinite number of rational cases possible.... – tatan Oct 27 '16 at 16:52
• Sorry. I misunderstood – Marvel Maharrnab Oct 27 '16 at 21:41
• Tatan now see the new proof is it ok – Marvel Maharrnab Oct 27 '16 at 22:02
• Yeah ok...but you should try to improve your answer by adding some more data on Fermat's Theorem...I will say your answer is correct but not well written... – tatan Oct 28 '16 at 3:28
• Brother see the link given by gerry myerson – Marvel Maharrnab Oct 28 '16 at 3:36