A) Centroid
B) Incentre
C) Circumcentre
D) Orthocentre (A rational point is a point both of whose coordinates are rational numbers)
Correct Answer: A
Solution :
If \[A=({{x}_{1}},\,\,{{y}_{1}}),\,\,B=({{x}_{2}},\,\,{{y}_{2}}),\,\,C=({{x}_{3}},\,\,{{y}_{3}}),\] where \[{{x}_{1}},\,\,{{y}_{1}},\] etc., are rational numbers then \[\Sigma {{x}_{1}},\,\,\Sigma {{y}_{1}}\] are also rational. So, the coordinates of the centroid \[\left( \frac{\Sigma {{x}_{1}}}{3},\,\,\frac{\Sigma {{y}_{1}}}{3} \right)\] will be rational. As \[AB=c=\sqrt{{{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}},}\,\,c\] may or may not be rational and it may be an irrational number of the form \[\sqrt{p}.\] Hence, the coordinates of the incentre \[\left( \frac{\Sigma a{{x}_{1}}}{\Sigma a},\,\,\frac{\Sigma a{{y}_{1}}}{\Sigma a} \right)\] may or may not be rational. If \[(\alpha ,\,\,\beta )\] be the circumcentre or orthocentre, a and b are found by solving two linear equations in \[\alpha ,\,\,\beta \] with rational coefficients. So \[\alpha ,\,\,\beta \]must be rational numbers.
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