1. **State the problem:** We are given a right triangle formed by points B, A, and C, where BA is the hypotenuse with length 35, one leg (from B to the foot of the perpendicular on BA) is 25, and the other leg is the radius $r$ we need to find. 2. **Formula used:** The Pythagorean theorem states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Assign values:** Let the radius $r$ be one leg, the other leg be 25, and the hypotenuse be 35. 4. **Apply the formula:** $$r^2 + 25^2 = 35^2$$ 5. **Calculate squares:** $$r^2 + 625 = 1225$$ 6. **Isolate $r^2$:** $$r^2 = 1225 - 625$$ $$r^2 = 600$$ 7. **Find $r$ by taking the square root:** $$r = \sqrt{600}$$ 8. **Simplify the square root:** $$r = \sqrt{100 \times 6} = \sqrt{100} \times \sqrt{6} = 10\sqrt{6}$$ **Final answer:** $$r = 10\sqrt{6}$$