1. **State the problem:** We have triangle PQRS with a segment QT parallel to RS. Given QT = 2x, PR = 15, TS = 12, and RS = x + 26, we want to find the value of $x$ using the properties of similar triangles. 2. **Formula and rules:** When a segment inside a triangle is parallel to one side, it creates similar triangles. The corresponding sides of similar triangles are proportional. Here, triangle PQT is similar to triangle PRS. 3. **Set up the proportion:** Since QT is parallel to RS, the ratio of corresponding sides is: $$\frac{QT}{RS} = \frac{PQ}{PR}$$ We know QT = 2x, RS = x + 26, and PR = 15. We need to find PQ or relate it to TS = 12. 4. **Use the segment TS:** Since QT is parallel to RS, TS corresponds to PT in the smaller triangle. The length TS = 12 is part of RS, so the ratio of QT to RS equals the ratio of PT to PR. 5. **Express the proportion:** $$\frac{QT}{RS} = \frac{PT}{PR}$$ Substitute values: $$\frac{2x}{x + 26} = \frac{PT}{15}$$ 6. **Relate PT and TS:** Since TS = 12 and RS = x + 26, PT = PR - TS = 15 - 12 = 3. 7. **Substitute PT = 3:** $$\frac{2x}{x + 26} = \frac{3}{15}$$ 8. **Simplify the right side:** $$\frac{3}{15} = \frac{1}{5}$$ 9. **Solve the equation:** $$\frac{2x}{x + 26} = \frac{1}{5}$$ Multiply both sides by $5(x + 26)$: $$5 \times 2x = 1 \times (x + 26)$$ $$10x = x + 26$$ 10. **Isolate $x$:** $$10x - x = 26$$ $$9x = 26$$ 11. **Divide both sides by 9:** $$x = \frac{26}{9}$$ 12. **Final answer:** $$x = \frac{26}{9} \approx 2.89$$