1. **Stating the problem:** We have a polygon with a rectangular base of 15 cm, two vertical sides of height $h$, and a top side divided into three segments: two angled segments of 7 cm each and a middle horizontal segment of 6 cm on each side (total 12 cm). The vertical sides adjacent to the base each measure 6 cm, and the total height from the base to the tips of the angled sides is $h$. We need to find the perimeter and solve for $h$. 2. **Understanding the perimeter:** The perimeter $P$ is the sum of all the sides of the polygon. 3. **List all sides:** - Base: 15 cm - Two vertical sides: each 6 cm - Two angled sides: each 7 cm - Top middle horizontal segment: 6 cm + 6 cm = 12 cm 4. **Calculate the perimeter:** $$P = 15 + 6 + 6 + 7 + 7 + 12$$ 5. **Simplify:** $$P = 15 + 12 + 14 + 12 = 53 \text{ cm}$$ 6. **Finding $h$:** The height $h$ is the vertical distance from the base to the tips of the angled sides. 7. **Use the Pythagorean theorem:** Each angled side of 7 cm forms a right triangle with vertical height $h - 6$ (since the vertical sides are 6 cm) and horizontal half of the top middle segment (6 cm). 8. **Set up the equation:** $$7^2 = (h - 6)^2 + 6^2$$ 9. **Calculate:** $$49 = (h - 6)^2 + 36$$ 10. **Isolate $(h - 6)^2$:** $$49 - 36 = (h - 6)^2$$ $$13 = (h - 6)^2$$ 11. **Take the square root:** $$h - 6 = \pm \sqrt{13}$$ 12. **Solve for $h$:** $$h = 6 \pm \sqrt{13}$$ 13. **Since height must be positive and greater than 6:** $$h = 6 + \sqrt{13} \approx 6 + 3.6055 = 9.6055 \text{ cm}$$ **Final answers:** - Perimeter $P = 53$ cm - Height $h \approx 9.61$ cm