Multiple choice

lf ABCDEF is a regular hexagon inscribed in a unit circle, then the product of lengths of lines AC and AE is

  1. 3
  2. ${\frac{3}{4}}$
  3. $3\sqrt{3}$
  4. $\displaystyle \frac{3\sqrt{3}}{2}$
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A Correct answer
Explanation

In a regular hexagon inscribed in a unit circle, the side length is 1. AC is a diagonal spanning two sides, length = 2 * sin(120/2) = sqrt(3). AE is also a diagonal spanning two sides, length = sqrt(3). Product = sqrt(3) * sqrt(3) = 3.

AI explanation

For a regular hexagon ABCDEF inscribed in a unit circle, each side subtends an angle of 60 degrees at the center. The length of a diagonal skipping one vertex, such as AC or AE, is given by the formula 2R sin(120 degrees / 2), which equals 2 * 1 * sin(60 degrees) = sqrt(3). The product of the lengths of lines AC and AE is therefore sqrt(3) * sqrt(3) = 3.

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