1. **State the problem:** We have two similar quadrilaterals CDEF and SRUT. Given the side lengths of CDEF: $CD=30$, $DE=63$, $EF=39$, $FC=81$, and some sides of SRUT: $TU=13$, $UR=21$, we need to find the lengths $ST$ and $RS$. 2. **Recall the property of similar polygons:** Corresponding sides of similar polygons are proportional. This means the ratio of any two corresponding sides in CDEF and SRUT is the same. 3. **Identify corresponding sides:** Since the quadrilaterals are named in order, the correspondence is: - $C \leftrightarrow S$ - $D \leftrightarrow R$ - $E \leftrightarrow U$ - $F \leftrightarrow T$ So, - $CD$ corresponds to $SR$ - $DE$ corresponds to $RU$ - $EF$ corresponds to $UT$ - $FC$ corresponds to $TS$ 4. **Set up ratios using known sides:** We know $DE=63$ corresponds to $RU=21$, so the scale factor $k$ from SRUT to CDEF is: $$k = \frac{DE}{RU} = \frac{63}{21} = 3$$ 5. **Find $ST$:** Since $FC=81$ corresponds to $TS=ST$, we have: $$ST = \frac{FC}{k} = \frac{81}{3} = 27$$ 6. **Find $RS$:** Since $CD=30$ corresponds to $SR=RS$, we have: $$RS = \frac{CD}{k} = \frac{30}{3} = 10$$ **Final answers:** $$ST = 27$$ $$RS = 10$$