A square circumscribes a circle and another square is inscribed in this circle with one vertex at the point of contact. The ratio of the areas of the circumscribed and the inscribed squares is

### Concept & Inscribed and Circumscribed Squares A square circumscribing a circle has a side length equal to the circle's diameter. A square inscribed in the same circle has a diagonal equal to the circle's diameter. ### Step-by-Step Solution * Let the radius of the circle be $r$. The diameter is $2r$. * For the circumscribed square (outer square): The side length is equal to the diameter of the circle, so $a_1 = 2r$. * Area of circumscribed square $A_1 = (2r)^2 = 4r^2$. * For the inscribed square (inner square): The diagonal is equal to the diameter of the circle, so $d = 2r$. * Area of inscribed square $A_2 = \frac{d^2}{2} = \frac{(2r)^2}{2} = \frac{4r^2}{2} = 2r^2$. * The ratio of the areas is $A_1 : A_2 = 4r^2 : 2r^2 = 2 : 1$. * Note: The phrase "with one vertex at the point of contact" just describes a specific rotation of the inner square, but does not affect its area. ### Exam Strategy & Shortcut Visualize the inner square rotated by 45 degrees. Its vertices hit the midpoints of the outer square's sides. Connecting these midpoints creates 4 triangles that fold perfectly into the center, meaning the inner square is exactly half the area of the outer square. The ratio is $2 : 1$. ### Common Pitfall Students may get confused by the detail "with one vertex at the point of contact," thinking it changes the geometry of the square. All squares inscribed in a given circle have the exact same area, regardless of rotation. ### Final Answer Therefore, the correct answer is **2 : 1**.