1. **Problem:** PQRS is an isosceles trapezoid with median MD. Given $SM = 4x + 5$ and $MP = 7x - 10$, find $RD$ and $DQ$. 2. **Formula and rules:** In an isosceles trapezoid, the median connects the midpoints of the legs and is parallel to the bases. The median divides the trapezoid into two equal segments on each leg, so $SM = RD$ and $MP = DQ$ because $M$ and $D$ are midpoints. 3. **Set equal segments:** $$SM = RD$$ $$MP = DQ$$ Since $SM = 4x + 5$ and $MP = 7x - 10$, and $RD = SM$, $DQ = MP$, we have: $$RD = 4x + 5$$ $$DQ = 7x - 10$$ 4. **Find $x$ using the trapezoid properties:** Since $RD$ and $DQ$ are parts of the trapezoid, and $RD$ and $DQ$ are segments on the base $PQ$, their sum equals $PQ$. But without $PQ$ given, we cannot find $x$ directly here. However, since $SM$ and $MP$ are parts of $SP$, and $SP$ is a leg, and $SM + MP = SP$, and $RD + DQ = PQ$, and $SP = PQ$ in isosceles trapezoid, we can set: $$SM + MP = RD + DQ$$ Substitute: $$ (4x + 5) + (7x - 10) = (4x + 5) + (7x - 10) $$ This is always true, so $x$ is not determined here. 5. **Final answer:** Since $SM = RD$ and $MP = DQ$, then: $$RD = 4x + 5$$ $$DQ = 7x - 10$$ Without more information, $RD$ and $DQ$ are expressed in terms of $x$ as above. --- **Summary:** - $RD = 4x + 5$ - $DQ = 7x - 10$