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Options

A. centroid
B. incentre
C. circumentre
D. orthoentre

Answer

B

Explanation

Let `A(x_(1),y_(1)), B(x_(2),y_(2)) and C(x_(3),y_(3))` be the vertices of `Delta ABC` such that `x_(1),x_(2),x_(3),y_(1),y_(2),y_(3)` are rational numbers, Then,
`(x_(1)+x_(2)+x_(3))/(3) and (y_(1)+y_(2)+y_(3))/(3)` are alo rational numbers.
So, the coordinates of the centroid are always rational numbers.
Let P{ bhe the circumcentre of `Delta ABC.` then,
`PA=AB=PC`
`rArr PA^(2)=PB^(2)=PC^(2)`
`rArr PA^(2)=PB^(2) and PB^(2)=PC^(2)`
Theses two relations two linear equations in terms of the coordiants of point P such that the coefficients are rational numbers. So, the coordinates of P are also rational numbers. Since the centroid G divides the segment joining circumcentre P and orthocentre H in the ratio of `1:2` Therefore, co-ordinates of H will have rational values.
The coordinates of the incentre are
`((ax_(1)+bx_(2)+cx_(3))/(a+b+c),(ay_(1)+by_(2)+cy+_(3))/(a+b+c))`
where `a=Bc,b=CA and c=AB`
Clearly, `a=sqrt((x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2))`
`b=sqrt((x_(3)-x_(1))^(2)+(y_(2)-y_(3))^(2))`
may not be ratioal numbers. Therefore, coordinats of the incentre may not be rational numbers.