Coordinate Geometry Question 14
Question: If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) always rational point(s)
[IIT 1998]
Options:
A) Centroid
B) Incentre
C) Circumcentre
D) Orthocentre (A rational point is a point both of whose coordinates are rational numbers)
Show Answer
Answer:
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 $ ( \frac{\Sigma x_1}{3},\frac{\Sigma y_1}{3} ) $ 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 $ ( \frac{\Sigma ax_1}{\Sigma a},\frac{\Sigma ay_1}{\Sigma a} ) $ 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.
BETA
