Is this possible(pythagorean triples)

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..

• 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 (?) – sedrick Mar 31 at 7:42
• Im not restricted to integers, but thought it would be something easier since the other problems from the list were super easy.. – user765760 Mar 31 at 8:17
• @sedrick Also, thanks! – user765760 Mar 31 at 8:17
• 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 $(6,10,11.66190379)$ – poetasis Sep 25 at 21:14

1 Answer

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.