1. **Problem statement:** Given trapezoid PQRS with sides $PQ=3$, $SR=9$, legs $PS=4$, $QR=6$, and a smaller trapezoid LMQP inside it such that $LM \parallel SR \parallel PQ$. The perimeters of trapezoids LMRS and LMQP are equal. We need to find the ratio $\frac{LS}{PL}$. 2. **Known values:** - $PQ=3$, $SR=9$ - $PS=4$, $QR=6$ - $LM \parallel SR \parallel PQ$ - Perimeter of $LMRS = $ Perimeter of $LMQP$ 3. **Express perimeters:** - Perimeter of $LMRS = LM + MR + RS + SL$ - Perimeter of $LMQP = LM + MQ + QP + PL$ 4. Since $LM \parallel SR \parallel PQ$, trapezoids $LMRS$ and $LMQP$ share side $LM$. Also, $RS = SR = 9$ and $PQ = 3$. 5. Let $LS = x$ and $PL = y$. We want to find $\frac{x}{y}$. 6. Using the perimeter equality: $$LM + MR + RS + SL = LM + MQ + QP + PL$$ Cancel $LM$ on both sides: $$MR + RS + SL = MQ + QP + PL$$ Substitute known lengths: $$MR + 9 + x = MQ + 3 + y$$ 7. Since $LM \parallel SR \parallel PQ$, the trapezoids are similar, so the legs are proportional: $$\frac{LS}{PL} = \frac{MR}{MQ} = k$$ 8. Rewrite the perimeter equality using $k$: $$MR + 9 + x = MQ + 3 + y$$ Substitute $MR = k MQ$ and $x = k y$: $$k MQ + 9 + k y = MQ + 3 + y$$ Group terms: $$k MQ - MQ + k y - y = 3 - 9$$ $$MQ(k - 1) + y(k - 1) = -6$$ Factor: $$(k - 1)(MQ + y) = -6$$ 9. Since lengths are positive, $MQ + y > 0$, so for the product to be negative, $k - 1 < 0$, thus $k < 1$. 10. Using the legs $PS=4$ and $QR=6$, and similarity ratios, the ratio $k = \frac{LS}{PL}$ corresponds to the ratio of legs, so: $$k = \frac{LS}{PL} = \frac{3}{4}$$ **Final answer:** $$\boxed{\frac{LS}{PL} = \frac{3}{4}}$$