1. **State the problem:** We have an isosceles trapezoid DEFG with sides DF and EG given by expressions $DF = 10s + 1$ and $EG = 7s + 40$. We need to find the value of $s$. 2. **Recall the property of isosceles trapezoids:** In an isosceles trapezoid, the non-parallel sides (legs) are equal in length. Here, DF and EG are the legs, so: $$DF = EG$$ 3. **Set up the equation:** $$10s + 1 = 7s + 40$$ 4. **Solve for $s$:** Subtract $7s$ from both sides: $$10s + 1 - \cancel{7s} = \cancel{7s} + 40 - 7s$$ $$3s + 1 = 40$$ Subtract 1 from both sides: $$3s + 1 - 1 = 40 - 1$$ $$3s = 39$$ Divide both sides by 3: $$\frac{3s}{\cancel{3}} = \frac{39}{\cancel{3}}$$ $$s = 13$$ 5. **Final answer:** $$\boxed{13}$$