1. **Problem Statement:** We have an isosceles trapezium PQRS with lines TU, PQ, and SR parallel. Given the ratio $ST : TP = 3 : 4$ and the length $RQ = 42$ cm, we need to find the lengths of $RU$ and $UQ$. 2. **Understanding the figure and notation:** Since PQRS is an isosceles trapezium, $PQ \parallel SR$ and $PQ = SR$ in length. TU is also parallel to PQ and SR, so $TU \parallel PQ \parallel SR$. 3. **Using the ratio $ST : TP = 3 : 4$:** This means that point T divides segment SP into parts such that $ST = 3k$ and $TP = 4k$ for some $k$. 4. **Given $RQ = 42$ cm:** Since RQ is a segment on the trapezium, and TU is parallel to PQ and SR, we can use properties of similar triangles formed by these parallel lines. 5. **Finding $RU$ and $UQ$:** Because TU is parallel to PQ and SR, triangles formed by points R, U, Q and points S, T, P are similar. The ratio of segments on these triangles corresponds to the ratio of the sides. 6. **Calculate $RU$ and $UQ$ using the ratio:** Since $ST : TP = 3 : 4$, the total length $SP = ST + TP = 7k$. The segment $RQ$ corresponds to the entire base length of 42 cm. 7. **Using similarity, $RU$ corresponds to $ST$ and $UQ$ corresponds to $TP$:** $$RU = \frac{3}{7} \times 42 = 18 \text{ cm}$$ $$UQ = \frac{4}{7} \times 42 = 24 \text{ cm}$$ **Final answer:** $RU = 18$ cm and $UQ = 24$ cm.