A circle was inscribed into the ABC triangle, and thus the tangent segments equal to 3 cm, 5 cm and 12 cm were obtained. Is the ABC triangle rectangular?
Let’s construct the radii OH, OM and OK to the points of tangency of the circle and the sides of the triangle.
The radii drawn to the tangent points are perpendicular to the tangents themselves.
By the property of tangents drawn from one point, BK = BM = 3 cm, AK = AH = 5 cm, CM = CH = 12 cm.
Then AB = AK + BK = 5 + 3 = 8 cm, BC = BM + CM = 3 + 12 = 15 cm, AC = AH + CH = 5 + 12 = 17 cm.
AC ^ 2 = 17 ^ 2 = 289.
AB + BC = 8 ^ 2 + 15 ^ 2 = 64 + 226 = 289.
289 = 289, therefore triangle ABC is rectangular.
Answer: Yes, the triangle is rectangular.
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