Question

The x-coordiante of the incentre of the triangle that has the coordiantes of mid points of its sides as (0,1),(1,1) and (1,0) is:
A. `2+sqrt(2)`
B. `2-sqrt(2)`
C. `1+sqrt(2)`
D. `1-sqrt(2)`

Answer 1

Correct Answer - B
If `D(x_(1),y_(1),E(x_(2),y_(2)) and F(x_(3),y_(3))` are mid-points of the sides BC,AC and AB respectively of `Delta ABC`. Then, the coordinates of vertices are `A(x_(2)+x_(3)-x_(1),y_(2)+y_(3)-y_(1)), B(x_(3)+x_(1)-x_(2),y_(3)+y_(1)-y_(2)) and C(x_(1)+x_(2)-x_(3),y_(1)+y_(2),y_(3))`

In the given problem the coordinates of the mid-points are D(0,1),E(1,1) and F((1,0). so, the coordinates of vertices are A((2,0) B(0,0) and C(0,2).
`:. AB=2, BC=2 and AC=2sqrt(2)`
Hence, the coordinats of the incentre are
`((2xx2xx+2sqrt(2)xx0+2xx0)/(2+2sqrt(2)+2),(2xx0+2sqrt(2)xx0+2xx2)/(2+2sqrt(2)+2))`
`((2)/(2+sqrt(2)),(2)/(2+sqrt(2))) or (2-sqrt(2),2-sqrt(2))`