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
\(\sqrt{15}\)
\(\sqrt{13}\)
\(\sqrt{12}\)
\(\sqrt{14}\)
The Correct Option is B
Solution and Explanation
Considering a regular hexagon as made of 6 equilateral triangles, a line connecting the farthest vertices of the hexagon can be seen as formed by the sides of two opposite equilateral triangles within the hexagon. Consequently, its length would be twice the length of the side of the hexagon, which in this case is 4 cm.
Given that line \( AD \) divides the hexagon into two symmetrical halves, it bisects angle \( D \), thereby establishing angle \( ADC \) as 60°.
The value of \( AT \) can be determined using the cosine formula:
\[ AT^2 = 4^2 + 1^2 - 2 \times 1 \times 4 \times \cos 60^\circ \] Simplifying the equation: \[ AT^2 = 16 + 1 - 8 \times \frac{1}{2} \] \[ AT^2 = 17 - 4 \] \[ AT^2 = 13 \] Taking the square root of both sides: \[ AT = \sqrt{13} \]
Final Answer:
The correct option is \( \boxed{(B): \sqrt{13}} \).
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