Question:medium

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. It T is the mid point of CD, then the length of AT, in cm, is

Updated On: Jan 15, 2026

  • \(\sqrt{15}\)

  • \(\sqrt{13}\)

  • \(\sqrt{12}\)

  • \(\sqrt{14}\)

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The Correct Option is B

Solution and Explanation

A regular hexagon can be conceptually divided into 6 equilateral triangles. A line connecting two opposite vertices of the hexagon is formed by the sides of two opposing equilateral triangles. Thus, its length is double the side length of the hexagon, which is 4 cm.

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm.

Line \( AD \) divides the hexagon into two equal halves and bisects angle \( D \), resulting in angle \( ADC \) being 60°.

The cosine formula is used to calculate \( AT \):

\[ AT^2 = 4^2 + 1^2 - 2 \times 1 \times 4 \times \cos 60^\circ \] Simplifying the equation yields: \[ AT^2 = 16 + 1 - 8 \times \frac{1}{2} \] \[ AT^2 = 17 - 4 \] \[ AT^2 = 13 \] Taking the square root of both sides gives: \[ AT = \sqrt{13} \]

Final Answer:

The correct option is \( \boxed{(B): \sqrt{13}} \).

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