1. **State the problem:** We need to find the value of $x$, the length of the top base of a trapezoid, given the bottom base is 31 and the two non-parallel sides (legs) are 17 and 19. 2. **Identify the formula:** The trapezoid has two legs and two bases. We can use the Pythagorean theorem on the right triangles formed by dropping perpendiculars from the top base to the bottom base. 3. **Set up variables:** Let the trapezoid have bottom base $b = 31$, top base $x$, and legs $a = 17$ and $c = 19$. Let the heights from the top base to the bottom base be $h$ (same for both since perpendiculars are shown). 4. **Express the segments:** The bottom base is split into three parts: the top base $x$ in the middle, and two segments on the sides, say $d$ and $e$, such that $d + x + e = 31$. 5. **Use Pythagorean theorem:** For the left triangle with leg 17 and height $h$, the horizontal segment is $d = \sqrt{17^2 - h^2}$. For the right triangle with leg 19 and height $h$, the horizontal segment is $e = \sqrt{19^2 - h^2}$. 6. **Write the equation:** $$d + x + e = 31 \implies \sqrt{289 - h^2} + x + \sqrt{361 - h^2} = 31$$ 7. **Isolate $x$:** $$x = 31 - \sqrt{289 - h^2} - \sqrt{361 - h^2}$$ 8. **Find $h$:** The height $h$ is the same for both triangles, so we can find $h$ by solving the system. But we have two unknowns $x$ and $h$. However, the problem states the heights are 17 and 19 on the legs, which are the legs themselves, so the height is unknown. 9. **Use the fact that the trapezoid is right-angled at the heights:** The height $h$ is the same for both triangles, so we can solve for $h$ by setting the sum of horizontal segments plus $x$ equal to 31. 10. **Try to find $h$ numerically:** Let’s try to find $h$ such that $$\sqrt{289 - h^2} + \sqrt{361 - h^2} + x = 31$$ Since $x$ is the top base, and the trapezoid is symmetric, we can approximate $h$ by trial or use algebraic methods. 11. **Alternatively, use the formula for the height of trapezoid with legs $a$, $c$, bases $b$, $x$:** $$h = \sqrt{a^2 - d^2} = \sqrt{c^2 - e^2}$$ But since $d + x + e = b$, and $d = \sqrt{a^2 - h^2}$, $e = \sqrt{c^2 - h^2}$, we can solve for $h$ numerically. 12. **Numerical solution:** Try $h = 12$: $$\sqrt{289 - 144} = \sqrt{145} \approx 12.04$$ $$\sqrt{361 - 144} = \sqrt{217} \approx 14.73$$ Sum: $12.04 + 14.73 = 26.77$, so $x = 31 - 26.77 = 4.23$ Try $h = 13$: $$\sqrt{289 - 169} = \sqrt{120} \approx 10.95$$ $$\sqrt{361 - 169} = \sqrt{192} \approx 13.86$$ Sum: $10.95 + 13.86 = 24.81$, so $x = 31 - 24.81 = 6.19$ Try $h = 11$: $$\sqrt{289 - 121} = \sqrt{168} \approx 12.96$$ $$\sqrt{361 - 121} = \sqrt{240} \approx 15.49$$ Sum: $12.96 + 15.49 = 28.45$, so $x = 31 - 28.45 = 2.55$ Try $h = 12.5$: $$\sqrt{289 - 156.25} = \sqrt{132.75} \approx 11.52$$ $$\sqrt{361 - 156.25} = \sqrt{204.75} \approx 14.31$$ Sum: $11.52 + 14.31 = 25.83$, so $x = 31 - 25.83 = 5.17$ 13. **Estimate $x$ around 5.2:** The value of $x$ is approximately 5.2 when rounded to the nearest tenth. **Final answer:** $$\boxed{5.2}$$