1. **State the problem:** We have a right triangle with one leg of length 30, hypotenuse 40, and the other leg divided into two segments: one segment is $x$ (vertical segment), and the base is 32. We need to find the value of $x$. 2. **Identify the right triangles:** The vertical segment $x$ divides the base into two parts, creating two right triangles inside the larger triangle. 3. **Use the Pythagorean theorem:** For the larger triangle, the legs are 30 and $x+32$, and the hypotenuse is 40. The Pythagorean theorem states: $$30^2 + (x+32)^2 = 40^2$$ 4. **Calculate squares:** $$900 + (x+32)^2 = 1600$$ 5. **Isolate the squared term:** $$(x+32)^2 = 1600 - 900 = 700$$ 6. **Take the square root:** $$x + 32 = \sqrt{700} = \sqrt{100 \times 7} = 10\sqrt{7}$$ 7. **Solve for $x$:** $$x = 10\sqrt{7} - 32$$ 8. **Approximate $\sqrt{7} \approx 2.64575$:** $$x \approx 10 \times 2.64575 - 32 = 26.4575 - 32 = -5.5425$$ Since $x$ cannot be negative, this suggests a misinterpretation. Instead, consider the smaller right triangle with legs $x$ and 30, and hypotenuse 40. 9. **Apply Pythagorean theorem to smaller triangle:** $$x^2 + 30^2 = 40^2$$ 10. **Calculate:** $$x^2 + 900 = 1600$$ 11. **Isolate $x^2$:** $$x^2 = 1600 - 900 = 700$$ 12. **Take square root:** $$x = \sqrt{700} = 10\sqrt{7} \approx 26.4575$$ 13. **Check options:** The closest option to 26.4575 is 28. **Final answer:** $\boxed{28}$