Just an ordinary triangle
Geometry
Level
2
(PMO 2017) If one of the legs of a right triangle has length 17 and the lengths of the other two sides are integers, then what is the radius of the circle inscribed in that triangle?
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1 solution
If 17 is one leg of the right triangle, and the lengths of other sides are integers too, 17 must be a part of a pythagoricien triple .
There is only one such triple including 17 : 17, 144, 145 .
Then the area of the triangle is 17 x 144 / 2 = 1,224. The perimeter is 17 + 144 +145 = 306.
The radius of the inscribed circle is : Area x 2 / Perimeter, i. e. 1,224 x 2 / 306 = 8 .