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

We need to find the distance AT where T is the midpoint of side CD in regular hexagon ABCDEF with side length 2 cm.

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Let us place the regular hexagon in a coordinate system to make calculations easier. We'll put the center at the origin and vertex A on the positive x-axis.

Key insight: In a regular hexagon with side length 2, the distance from center to any vertex is also 2. This is because each "slice" of the hexagon (connecting center to adjacent vertices) forms an equilateral triangle.

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With center at origin and A at (2, 0), the vertices are:

A: (2, 0)

B: (1, √3)

C: (-1, √3)

D: (-2, 0)

E: (-1, -√3)

F: (1, -√3)

In a regular hexagon, vertices are separated by 60° angles. Using trigonometry:

B is at angle 60°: $(2\cos60°, 2\sin60°) = (1, √3)$

C is at angle 120°: $(2\cos120°, 2\sin120°) = (-1, √3)$

D is at angle 180°: $(-2, 0)$

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Since T is the midpoint of CD:

C is at $(-1, √3)$

D is at $(-2, 0)$

T is at $\left(\dfrac{-1 + (-2)}{2}, \dfrac{√3 + 0}{2}\right) = \left(-\dfrac{3}{2}, \dfrac{√3}{2}\right)$

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Now we can find the distance AT using the distance formula:

$AT = \sqrt{\left(2 - \left(-\dfrac{3}{2}\right)\right)^2 + \left(0 - \dfrac{√3}{2}\right)^2}$

$AT = \sqrt{\left(2 + \dfrac{3}{2}\right)^2 + \left(-\dfrac{√3}{2}\right)^2}$

$AT = \sqrt{\left(\dfrac{7}{2}\right)^2 + \left(\dfrac{√3}{2}\right)^2}$

$AT = \sqrt{\dfrac{49}{4} + \dfrac{3}{4}}$

$AT = \sqrt{\dfrac{52}{4}}$

$AT = √13$

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The reference solution uses a clever geometric approach:

Find AC first: AC is a "long diagonal" of the hexagon. In the coordinate system, AC goes from $(2, 0)$ to $(-1, √3)$, so $AC = \sqrt{(2-(-1))^2 + (0-√3)^2} = \sqrt{9 + 3} = \sqrt{12} = 2√3$

Use right triangle: The reference creates a right triangle where one leg is the horizontal distance from A to the vertical line through T, the other leg is the vertical distance from that point to T, and the hypotenuse is AT.

Apply Pythagorean theorem: $AT^2 = (2√3)^2 + (1)^2 = 12 + 1 = 13$

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Therefore, $AT = √13$ cm.

Key takeaway: Both coordinate geometry and the geometric construction approach give the same answer, but the coordinate method might be more intuitive for beginners.