Multiple choice

If in a triangle $A = ( 1,10 ) ,$ circumcentre $= \left( - \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ and orhocentre $= \left( \frac { 11 } { 3 } , \frac { 4 } { 3 } \right)$ then the co-ordinate of mid-point of side opposite to $A$ is

  1. $( 1,6 )$
  2. $( 1,5 )$
  3. $( 1 , - 3 )$
  4. $( 1 , - \dfrac { 11 } { 3 })$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In any triangle, the centroid G divides the segment joining the orthocenter H and circumcenter O in the ratio 2:1. G = (2O + H)/3 = (2(-1/3, 2/3) + (11/3, 4/3))/3 = ((-2/3+11/3)/3, (4/3+4/3)/3) = (3/3, 8/9) = (1, 8/9). The midpoint M of the side opposite to A satisfies (A + 2M)/3 = G. (1, 10) + 2M = 3(1, 8/9) = (3, 8/3). 2M = (2, -22/3), so M = (1, -11/3).

AI explanation

The vertices of a triangle, its orthocentre, and its circumcentre are connected by Euler's line, where the centroid divides the line segment from the orthocentre to the circumcentre in a 2:1 ratio. Using the section formula with orthocentre (11/3, 4/3) and circumcentre (-1/3, 2/3), the centroid G is calculated as ((11/3 - 2/3)/3, (4/3 + 4/9)/3) = (1, 16/9). In any triangle, the centroid also divides the median from a vertex to the midpoint of the opposite side in a 2:1 ratio, so we set up the equation G = ((2D + A)/3) using vertex A(1,10) to find midpoint D. Solving (1, 16/9) = ((2x + 1)/3, (2y + 10)/3) results in D = (1, -11/3).

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