In a circle of radius r, an equilateral triangle is inscribed. Express the side length s in terms of r.

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

In a circle of radius r, an equilateral triangle is inscribed. Express the side length s in terms of r.

Explanation:
In an inscribed equilateral triangle, the circle’s radius is the circumradius, and each side is a chord that subtends a central angle of 120 degrees. Using the chord-length relation, the side length s equals 2r sin(120°/2) = 2r sin(60°) = 2r · (√3/2) = √3 r. So the side length grows with the circle radius by a factor of √3.

In an inscribed equilateral triangle, the circle’s radius is the circumradius, and each side is a chord that subtends a central angle of 120 degrees. Using the chord-length relation, the side length s equals 2r sin(120°/2) = 2r sin(60°) = 2r · (√3/2) = √3 r. So the side length grows with the circle radius by a factor of √3.

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