Multiple choice

In a triangle $ABC,\,\angle{A}={30}^{\circ},\,\,H$ is the orthocentre and $D$ is the midpoint of $BC$.Segment $HD$ is produced to $T$ such that $HD=DT$.The ratio of the lengths $\dfrac{AT}{BC}$ is equal to

  1. $1:2$
  2. $2:1$
  3. $3:2$
  4. $2:3$
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B Correct answer
Explanation

In a triangle, the distance from the orthocenter to a vertex is 2R cos A. The segment HD produced to T such that HD=DT makes AT a diameter or related to circumradius. The ratio AT/BC simplifies to 2:1.

AI explanation

By the property of the Euler line and the nine-point circle, point T is the reflection of the orthocentre H across the midpoint D of BC, making it a point on the circumcircle exactly opposite to vertex A, so AT is the diameter of the circumcircle. Using the extended sine rule, BC = 2R sin A = 2R sin(30) = R. Since AT is the diameter of the circumcircle, AT = 2R, making the ratio AT to BC equal to 2R to R, which is 2:1.

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