Babylonian Mathematics/Examples/Pythagorean Triangle whose Side Ratio is 1.54

Example of Babylonian Mathematics
Consider a Pythagorean triangle whose hypotenuse and one leg are in the ratio $1.54 : 1$.

What are the lengths of that hypotenuse and that leg?

Solution
The lengths in question are $829$ and $540$.

Proof
Let $a$, $b$ and $c$ be positive integers such that $a^2 + b^2 = c^2$ and such that $1.54 \times a = c$.

, suppose $a$ is even.

From Solutions of Pythagorean Equation, there exist positive integers $p$ and $q$ such that:

Hence it follows that:
 * $\dfrac c a = \dfrac 1 2 \paren {\dfrac p q + \dfrac q p}$

The Babylonians would then consult the various standard tables of reciprocals which they used for multiplication.

Without these tables, we set:
 * $\dfrac p q = t$

and solve the quadratic equation:

We can discard $0.369$ because we are after $p > q$.

Hence:

which is what is found in the original Babylonian clay tablet.