1. **State the problem:** Calculate the perimeter of quadrilateral ABCD, which is circumscribed around a circle. 2. **Important property:** For a quadrilateral circumscribed around a circle (a tangential quadrilateral), the sum of the lengths of opposite sides are equal. That is, $$AB + CD = BC + DA$$. 3. **Given tangent lengths:** The tangents from each vertex to the points of tangency with the circle are: - From A: 4 cm - From B: 5 cm - From C: 3 cm - From D: 7 cm 4. **Tangent length property:** Tangents drawn from an external point to a circle are equal in length. So, the two tangents from each vertex to the circle are equal. 5. **Assign tangent segments:** Let the points of tangency on sides be such that: - At vertex A, tangents are both 4 cm - At vertex B, tangents are both 5 cm - At vertex C, tangents are both 3 cm - At vertex D, tangents are both 7 cm 6. **Calculate side lengths:** Each side length is the sum of the two tangent segments from its endpoints. - Side AB = tangent from A + tangent from B = 4 + 5 = 9 cm - Side BC = tangent from B + tangent from C = 5 + 3 = 8 cm - Side CD = tangent from C + tangent from D = 3 + 7 = 10 cm - Side DA = tangent from D + tangent from A = 7 + 4 = 11 cm 7. **Calculate perimeter:** $$\text{Perimeter} = AB + BC + CD + DA = 9 + 8 + 10 + 11 = 38 \text{ cm}$$ 8. **Justification:** The perimeter is the sum of all sides calculated using tangent lengths, consistent with the property of tangential quadrilaterals. **Final answer:** $$\boxed{38 \text{ cm}}$$