A3. From the data in the problem, we calculate $\Varangle B A T=30^{\circ}$. Since $\tan (\Varangle B A T)=\frac{|T B|}{|A B|}$, it follows that $|T B|=|A B| \tan 30^{\circ}=1 \cdot \frac{\sqrt{3}}{3}=\frac{\sqrt{3}}{3}$. The correct answer is $D$.

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A3. The angle bisectors from vertex $A$ of a unit square $ABCD$ divide the right angle into three equal angles. One of the angle bisectors intersects side $BC$ at point $T$. What is the length of segment $BT$?
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{1}{3}$
(D) $\frac{\sqrt{3}}{3}$
(E) $\frac{\sqrt{2}}{2}$