1. **State the problem:** We need to find the lengths of the parallel sides of a trapezium given its area, the ratio of the parallel sides, and the perpendicular distance between them. 2. **Given:** - Area $A = 340$ cm$^2$ - Ratio of parallel sides $= 10:7$ - Height $h = 5$ cm 3. **Formula for the area of a trapezium:** $$A = \frac{1}{2} \times (a + b) \times h$$ where $a$ and $b$ are the lengths of the parallel sides. 4. **Express the sides in terms of a variable:** Let the common ratio factor be $x$. Then, $$a = 10x, \quad b = 7x$$ 5. **Substitute into the area formula:** $$340 = \frac{1}{2} \times (10x + 7x) \times 5$$ 6. **Simplify:** $$340 = \frac{1}{2} \times 17x \times 5 = \frac{1}{2} \times 85x = 42.5x$$ 7. **Solve for $x$:** $$x = \frac{340}{42.5} = 8$$ 8. **Find the lengths of the parallel sides:** $$a = 10x = 10 \times 8 = 80 \text{ cm}$$ $$b = 7x = 7 \times 8 = 56 \text{ cm}$$ **Final answer:** The lengths of the parallel sides are 80 cm and 56 cm.