Straight Line and Pair of Straight Lines 1 Question 22
22. 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
(2013 Main)
(a) $2+\sqrt{2}$
(b) $2-\sqrt{2}$
(c) $1+\sqrt{2}$
(d) $1-\sqrt{2}$
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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. $2 \sqrt{2}$.
$x$-coordinate of incentre $=\frac{2 \times 0+2 \sqrt{2} \cdot 0+2 \cdot 2}{2+2+2 \sqrt{2}}$
$$ =\frac{2}{2+\sqrt{2}} \times \frac{2-\sqrt{2}}{2-\sqrt{2}}=2-\sqrt{2} $$
