Multiple choice

In $\Delta ABC,\ D,\ E,\ F$ are midpoints of the sides $BC, CA$ and $AB$ respectively. $O$' is the circumcentre, $G$' is the centroid, $H$' is the orthocentre and $P$ is any point. Match the following List I List II $1) \vec{PA} +\vec{PB}+\vec{PC}$ $a) $$0$ $2)\vec {GA}+\vec{GB}+\vec{GC}$ $b) \vec{OH}$ $3)\displaystyle \vec{AD}+\dfrac{2}{3}\vec{BE}+\dfrac{1}{3}\vec{CF}$ $c)\vec{ PD}+\vec{PE}+\vec{PF}$ $4)\vec{OA}+\vec{OB}+\vec{OC}$ $d){\displaystyle\dfrac{1}{2}}\vec{AC}$

  1. $1-a,2- b,3- c,4- d$
  2. $1-c,2- a,3- b,4- c$
  3. $1-c,2- a,3- d,4- b$
  4. $1-a,2- b,3- d,4- c$
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C Correct answer
Explanation

1) Sum of vectors from point P to vertices is 3 * vector from P to centroid G. 2) Sum of vectors from centroid to vertices is 0. 3) AD + 2/3 BE + 1/3 CF is a vector identity related to sides. 4) Sum of vectors from circumcenter to vertices is OH.

AI explanation

Using the midpoint rule, vector PA plus vector PB plus vector PC equals 3 times vector PG, which can be rewritten as vector PD plus vector PE plus vector PF. This matches 1 with c. By the definition of a centroid, the sum of the vectors from the centroid to the vertices GA plus GB plus GC is the zero vector, which matches 2 with a. The vector sum AD plus (2/3)BE plus (1/3)CF simplifies algebraically to 1/2 times vector AC, matching 3 with d. Since O is the circumcentre, the vector sum OA plus OB plus OC equals OH, matching 4 with b. The correct sequence is 1-c, 2-a, 3-d, 4-b.

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