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In right triangle ABC, the right angle is at B, and BD is the altitude from B to hypotenuse AC. Given BD = 9 cm, and the inradii of triangles ABC, ABD, and BCD are r1, r2, and r3 respectively, find r1 + r2 + r3.

1. 9
2. 18
3. 27
4. 6

Correct answer: 9

Solution

There is a beautiful theorem: if BD is the altitude from the right angle B to hypotenuse AC in right triangle ABC, then r_ABC + r_ABD + r_BCD = BD. This follows because for a right triangle with legs p, q and hypotenuse h, the inradius is (p + q - h)/2. Letting AD = m, DC = n, BD = h, AB = sqrt(mh_total... applying the identity carefully, the sum telescopes to BD = 9.

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