1. **State the problem:** We have a right triangle NSB with a right angle at N. Segment NW is perpendicular to the hypotenuse BS. Given NB = 12 units and WS = 16.1 units, we need to find the length BW. 2. **Understand the setup:** Since NW is perpendicular to the hypotenuse BS, point W lies on BS, splitting it into BW and WS. 3. **Use the property of right triangles with altitude to hypotenuse:** The altitude NW to the hypotenuse divides the hypotenuse into two segments BW and WS such that: $$NB^2 = BW \times BS$$ $$NS^2 = WS \times BS$$ 4. **Find BS:** Since W lies on BS, we have: $$BS = BW + WS$$ 5. **Express NB and NS:** We know NB = 12 units. We need NS to use the formula. Using the right triangle NSB with right angle at N, by Pythagoras: $$NS = \sqrt{BS^2 - NB^2}$$ 6. **Use the altitude property:** The altitude NW satisfies: $$NW^2 = BW \times WS$$ 7. **Calculate BW:** Since WS = 16.1, and NB = 12, we use the relation: $$NB^2 = BW \times (BW + WS)$$ $$12^2 = BW \times (BW + 16.1)$$ $$144 = BW^2 + 16.1 BW$$ 8. **Solve quadratic equation:** $$BW^2 + 16.1 BW - 144 = 0$$ Use quadratic formula: $$BW = \frac{-16.1 \pm \sqrt{16.1^2 + 4 \times 144}}{2}$$ Calculate discriminant: $$16.1^2 = 259.21$$ $$4 \times 144 = 576$$ $$\sqrt{259.21 + 576} = \sqrt{835.21} = 28.9$$ So: $$BW = \frac{-16.1 \pm 28.9}{2}$$ Choose positive root: $$BW = \frac{-16.1 + 28.9}{2} = \frac{12.8}{2} = 6.4$$ 9. **Final answer:** $$BW = 6.4$$ units (rounded to the nearest tenth).