Start by giving your students this challenge: Find the shaded area:

Here's the solution. Before attempting this challenge, students must know the congruent tangents theorem, and that there always exists a radius perpendicular to a tangent. Next, have your students generalize their solution to a right triangle with sides \(a,\) \(b,\) and \(c.\) They should find the inradius to be \((a + b - c) / 2.\) The reasoning is the same as that used to solve the challenge. After that, give your students this challenge: Find the radius of the semicircle:

Students are likely to arrive at *Solution 2.* Prereqs include knowing the hypotenuse-leg congruence theorem, knowing that there always exists a radius perpendicular to a tangent, and being able to solve similar triangles. Finally, have your students generalize for a right triangle with sides \(a,\) \(b,\) and \(c.\) The generalization requires exactly the same reasoning. Students should find the radius to be \(b(c - b) / a.\)