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  4. The x-coordinate of the incentre of the triangle that has the coordinates of mid-points of its sides as (0, 1), (1, 1) and (1,0) is

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Q. The $x$-coordinate of the incentre of the triangle that has the coordinates of mid-points of its sides as $(0, 1), (1, 1)$ and $(1,0)$ is

JEE MainJEE Main 2013Straight Lines

Solution:

Given mid-points of a triangle are $(0,1),(1,1)$ and $(1,0)$. Plotting these points on a graph paper and make a triangle.
So, the sides of a triangle will be $2,2$ and $\sqrt{2^{2}+2^{2}}$ i.e. $2 \sqrt{2}$
image
$x $ -coordinate of incentre $=\frac{2 \times 0+2 \sqrt{2} \cdot 0+2 \cdot 2}{2+2+2 \sqrt{2}}$
$=\frac{2}{2+\sqrt{2}} \times \frac{2-\sqrt{2}}{2-\sqrt{2}}=2-\sqrt{2} $