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rectangular cartensian co-ordinates — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSrectangular cartensian co-ordinates1 Mark
MCQ
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 $(s)$

(A rational ponit is a point both of whose coordinates are rational numbers)

  • A
    Centroid
  • B
    The length of the length
  • C
    Incentre
  • ✓
    All of above
✓

Answer

Correct option: D.
All of above
d
(d) 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, $\alpha$ and $\beta$ are found by solving two linear equations in $\alpha ,\,\,\beta $ with rational coefficients. So $\alpha ,\,\,\beta $must be rational numbers.

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