1. **State the problem:** We have a quadrilateral ABCD with an inscribed circle P tangent to all sides. Given sides AB = 12 units, DC = 9 units, AD = 4 units, and BC = 4 units, we need to find the perimeter of ABCD. 2. **Recall a property of tangential quadrilaterals:** For a quadrilateral with an inscribed circle (tangential quadrilateral), the sum of the lengths of the opposite sides are equal. That is, $$AB + CD = BC + AD$$ 3. **Check the given sides:** - AB = 12 - CD = 9 - AD = 4 - BC = 4 Calculate sums: $$AB + CD = 12 + 9 = 21$$ $$BC + AD = 4 + 4 = 8$$ These are not equal, which suggests a discrepancy in the problem statement or measurements. 4. **Re-examine the problem:** The problem states AB and DC are parallel and measure 12 units each, but also says CD measures 9 units. This is contradictory. Assuming the problem meant AB = 12, DC = 12, AD = 4, BC = 4, then: $$AB + CD = 12 + 12 = 24$$ $$BC + AD = 4 + 4 = 8$$ Still not equal. 5. **Alternative interpretation:** Since the problem states AB and DC are parallel and measure 12 units each, but also says CD measures 9 units, likely a typo. Let's assume AB = 12, DC = 9, AD = 4, BC = 4 as given. 6. **Use the property of tangential quadrilaterals:** The sums of opposite sides must be equal, so: $$AB + CD = BC + AD$$ Substitute known values: $$12 + 9 = 4 + 4$$ $$21 = 8$$ This is false. 7. **Conclusion:** The problem's given side lengths contradict the property of tangential quadrilaterals. However, since the problem asks for the perimeter, and all sides are given, the perimeter is simply the sum of all sides: $$Perimeter = AB + BC + CD + DA = 12 + 4 + 9 + 4 = 29$$ **Final answer:** $$\boxed{29}$$ units