3. In a regular hexagon $A B C D E F$ with side length $a$, the extensions of sides $A F$ and $D E$ intersect at point $T$. Calculate the exact length of segment $B T$.


3. From the sketch, we understand that $|D T|=2 a$, and $|B D|$ is equal to the length of two heights of an equilateral triangle with side $a$, thus $|B D|=a \sqrt{3}$. The length of the segment $B T$ is calculated using the Pythagorean theorem: $|B T|=\sqrt{|D T|^{2}+|B D|^{2}}=$ $\sqrt{4 a^{2}+3 a^{2}}=a \sqrt{7}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-09.jpg?height=440&width=576&top_left_y=1916&top_left_x=1254)

Sketch .1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-09.jpg?height=63&width=1636&top_left_y=2421&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-09.jpg?height=80&width=1642&top_left_y=2473&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-09.jpg?height=57&width=1639&top_left_y=2536&top_left_x=206)

Correct squaring of the relation $|B T|^{2}=(a \sqrt{3})^{2}+(2 a)^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_06886e410b39c0216094g-09.jpg?height=66&width=1639&top_left_y=2628&top_left_x=206)