Question:
In the figure below (not drawn to scale), rectangle ABCD is inscribed in the circle with center at O. The length of side AB is greater than side BC. The ratio of the area of the circle to the area of the rectangle ABCD is $\pi : \sqrt{3}$. The line segment DE intersects AB at E such that $\angle \text{DOC} = \angle \text{ADE}$. The ratio $AE : AD$ is: 
In the figure below (not drawn to scale), rectangle ABCD is inscribed in the circle with center at O. The length of side AB is greater than side BC. The ratio of the area of the circle to the area of the rectangle ABCD is $\pi : \sqrt{3}$. The line segment DE intersects AB at E such that $\angle \text{DOC} = \angle \text{ADE}$. The ratio $AE : AD$ is:
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In problems involving inscribed shapes, use the properties of similar triangles and circles to calculate the required ratios.
Updated On: Mar 23, 2026
- $\sqrt{3}$
- $\sqrt{2}$
- $2\sqrt{3}$
- $1 : 2$
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The Correct Option is B
Solution and Explanation
We are given that the rectangle ABCD is inscribed in a circle and the area of the circle is $\pi$ times the area of the rectangle. Since the diagonal of the rectangle is the diameter of the circle, we can calculate the ratio $AE : AD$ using geometric properties and similarity of triangles.
From the given condition $\angle \text{DOC} = \angle \text{ADE}$, we can use the properties of similar triangles to determine the ratio of the line segments.
Hence, the ratio of $AE : AD$ is $\sqrt{2}$.
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