1. **Problem 1:** Given an isosceles trapezoid ABCD with an inscribed circle of radius 6 and the difference of the bases $AD - BC = 9$, find the perimeter of ABCD. 2. **Formula and rules:** - For a trapezoid with an inscribed circle, the sum of the lengths of the legs equals the sum of the bases: $AB + CD = AD + BC$. - The radius $r$ of the inscribed circle relates to the trapezoid by the formula for the area $A = r imes s$, where $s$ is the semiperimeter. - The height $h$ of the trapezoid can be found using the radius and the trapezoid properties. 3. **Step 1: Define variables:** Let $AD = a$, $BC = b$, and the legs $AB = CD = c$ (since it is isosceles). Given $a - b = 9$ and $r = 6$. 4. **Step 2: Use the property of trapezoid with inscribed circle:** Since the trapezoid has an inscribed circle, the sum of the legs equals the sum of the bases: $$AB + CD = AD + BC \Rightarrow 2c = a + b$$ 5. **Step 3: Express perimeter $P$:** $$P = a + b + 2c$$ Using $2c = a + b$, then $$P = a + b + (a + b) = 2(a + b)$$ 6. **Step 4: Find height $h$ using radius $r$:** The height $h$ of the trapezoid equals the radius of the inscribed circle: $$h = r = 6$$ 7. **Step 5: Use Pythagoras theorem for leg $c$:** The legs are equal and can be found by: $$c = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2} = \sqrt{6^2 + \left(\frac{9}{2}\right)^2} = \sqrt{36 + \frac{81}{4}} = \sqrt{\frac{144}{4} + \frac{81}{4}} = \sqrt{\frac{225}{4}} = \frac{15}{2} = 7.5$$ 8. **Step 6: Use $2c = a + b$ to find $a + b$:** $$2c = a + b \Rightarrow 2 \times 7.5 = a + b \Rightarrow a + b = 15$$ 9. **Step 7: Calculate perimeter $P$:** $$P = 2(a + b) = 2 \times 15 = 30$$ 10. **Step 8: Check answer choices:** None of the options match 30 exactly, so re-examine step 6. 11. **Correction:** Actually, the perimeter is: $$P = a + b + 2c = (a + b) + 2c = (a + b) + (a + b) = 2(a + b)$$ Since $2c = a + b$, then $P = 2(a + b) = 2 \times 15 = 30$. 12. **But the problem's options do not include 30.** Reconsider the height and radius relation. 13. **Step 9: Relation between radius and trapezoid:** The radius $r$ of the inscribed circle in trapezoid is given by: $$r = \frac{h}{2}$$ where $h$ is the height. 14. **Step 10: Calculate height $h$:** $$h = 2r = 2 \times 6 = 12$$ 15. **Step 11: Recalculate leg $c$:** $$c = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2} = \sqrt{12^2 + \left(\frac{9}{2}\right)^2} = \sqrt{144 + 20.25} = \sqrt{164.25} = 12.82$$ 16. **Step 12: Use $2c = a + b$:** $$2 \times 12.82 = a + b = 25.64$$ 17. **Step 13: Calculate perimeter $P$:** $$P = a + b + 2c = 25.64 + 25.64 = 51.28$$ 18. **Step 14: Since $a - b = 9$, solve for $a$ and $b$:** $$a - b = 9$$ $$a + b = 25.64$$ Adding: $$2a = 34.64 \Rightarrow a = 17.32$$ Subtracting: $$2b = 16.64 \Rightarrow b = 8.32$$ 19. **Step 15: Final perimeter:** $$P = a + b + 2c = 17.32 + 8.32 + 2 \times 12.82 = 25.64 + 25.64 = 51.28$$ 20. **Step 16: Closest answer choice is 54 (D).** --- **Final answer:** D) 54