In a right triangle ABC the in circle touches the hypotenuse AC at O. If AD=10 and DC = 3, the inradius of ABC is

For a right triangle ABC with hypotenuse AC, the incircle touches the sides AB, BC, and AC at points that create equal tangent lengths from the vertices. We are given the tangent lengths on the hypotenuse as AD equals 10 and DC equals 3, making the total hypotenuse length 13. Let the inradius be r, which is also the length of the tangent from vertex B to the points of tangency on sides AB and BC. Using the right triangle area formula where the area equals r times the semi-perimeter, and also equals one-half the product of the legs, we set up r multiplied by 13 equals the product of the two unknown legs. By applying the Pythagorean theorem and substituting the tangent relationships to find the individual leg lengths, we solve for r to find it equals 2.

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