Garden Geometry 1588: Unlocking the Tangent: Prove the Fifth of the Radius in a Sector with Semicircle

Proving Tangency and Proportions in a Sector-Semicircle Configuration

Dive into the geometry of a circular sector combined with a semicircle. This problem challenges you to demonstrate a fascinating proportional relationship involving a tangent chord and the sector's radius. Analyze the setup, uncover the hidden connections, and prove that the length of the tangent segment is precisely one-fifth of the sector's radius. Perfect for geometry enthusiasts eager to explore elegant mathematical proofs.

Problem 1588 Statement

In a circular sector AOB with a 90-degree central angle, a semicircle centered at Q has OB as its diameter. The chord AD is tangent to the semicircle at C. Prove that CD is one-fifth of the sector's radius.

Uncover and share solutions to this problem.

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