Hexagon, from vertex to midpoint of a side. [closed]

geometry euclidean-geometry

${ABCDEF}$ is a regular hexagon with side length $2$ units, ${T}$ is the midpoint of side ${CD}$, what will be the length of the line ${AT}$?


Solution 1:

$ABCDEF$ is a regular hexagon, therefore, $AB=BC=CD=DE=EF=FA$ and all internal angles are equal to $120^{\circ}$. $T$ is the midpoint of side $CD$, therefore, $CT=DT$. $AB=2$.

$\triangle{ABC}$ - an isosceles triangle, $\implies$ $\angle{BAC}=\angle{BCA}=30^{\circ}$, $\implies$ $\angle{ACT}=90^{\circ}$ and $AC=2\sqrt{3}$, $\implies$ $AT=(AC^{2}+CT^{2})^{0.5}=\sqrt{13}$, $\implies$ $\bbox[lightgreen]{AT=\sqrt{13}}$.

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