My maths teacher gave me some problems and one of them was that I have a right triangle with side 6 and area 30.. So my questions are: 1.Is this even possible and 2.should I assume that most of the time, a right triangle with side 6 would be egyptian (pythagorean triple) P.S sorry if my sentence formation isn't top tier..

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    $\begingroup$ Yes, one leg is 6, so take the other side as 10 so the area is 30. I don't see why not. Unless the sides have to be integers (?) $\endgroup$ – sedrick Mar 31 at 7:42
  • $\begingroup$ Im not restricted to integers, but thought it would be something easier since the other problems from the list were super easy.. $\endgroup$ – user765760 Mar 31 at 8:17
  • $\begingroup$ @sedrick Also, thanks! $\endgroup$ – user765760 Mar 31 at 8:17
  • $\begingroup$ The only Pythagorean triple (integer sides required) with an area of $30$ is $(5,12,13)$. The only right triangle with one side of $6$ and an area of $30$ is $(6,10,11.66190379)$ $\endgroup$ – poetasis Sep 25 at 21:14

Two possibilities:

  1. The side of length 6 is a cathetus (adjacent to right angle): As the area is $30 = \frac12 c\cdot6$ where $c$ is the length of the other cathetus, there must be $c=10$.

  2. The side of length 6 is the hypothenuse (opposite of right angle): The area is $30=\frac12h\cdot6$ where $h$ is the height of the triangle, thus $h = 10$. Moreover, as it is a right triangle, the right angle is at Thales' circle which has radius $6/2 = 3$ around the middle of the hypothenuse. As this circle does not intersect a line that is $h=10$ units away from the hypothenuse and parallel to it, no solutions from this case.

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