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# In a , H is the orthocentre and D is the mid point of Segment HD is produced to T such that HD=DT. Then the ratio equals to ;

- 1:2
- 2:1
- 3:2
- 2:3

## The correct answer is: 2:1

### We should find the ratio of AT to BC using all the given data

Let us assume that the circumcenter of $△ABC$ is O and is situated at the origin of the coordinate plane. Hence, the vector notation for all the vertices is
$OA=a$
$OB=b$
$OC=c$
Also, as O is the circumcenter of $△ABC$,
$a=b=c=R$ (circum-radius)
Since D is mid-point of $BC$, $d=2b+c $
$∴a+b+c=a+2d$
Also, $2OD=AH$ as H is the orthocentre and it divides height in 2:1 ratio.
$∴a+b+c=a+2d=OA+AH=OH=h…(1)$
According to given information, $HD=DT$
$⟹2b+c =2h+t $
$⟹2b+c =2a+b+c+t $
$⟹a=−t$
$⟹AT=2a$
$⟹∣AT∣=2∣a∣=2R$
Also, from property of side-length of triangle,
$BC=2RsinA=R$ ($∵∠A=30_{∘}$)
$∴AT=2BC$
AT/BC=2:1

Hence 2:1 is the correct option

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