In triangle $ABC$, $\angle A = 30^\circ$, $H$ is the orthocentre and $D$ is the midpoint of $BC$. Segment $HD$ is produced such that $HD = DT$. The length $AT$ equals:
a) $2 BC$
b) $3BC$
c) $\dfrac{4}{3}BC$
d) none of these
Attempt:
Let $H$ be the origin. Then define $\vec a$, $\vec b$ , $\vec c$ directed towards vertices A, B and C respectively.
Given: $\vec d = \dfrac{\vec b + \vec c}{2}$
Therefore $\vec T$ directed towards point T from H = $\vec b+ \vec c$
$\implies \vec{AT} = \vec T - \vec A = \vec b + \vec c - \vec a$
Now I am facing trouble in finding the magntiude of $\vec b + \vec c - \vec a$.
How do I continue from here?