First slide

Cartesian plane

Question

The x-coordinates of the incentre of the triangle where the mid-points of the sides are $(0, 1), (1, 1)$ and $(1, 0)$ is

Moderate

A

$2 + \sqrt{2}$
B

$1 + \sqrt{2}$
C

$2 - \sqrt{2}$
D

$1 - \sqrt{2}$

Solution

Let $D (0, 1), E (1, 1)$ and $F (1, 0)$ be the mid-points
of sides $B C, C A and A B$ respectively of $△ A B C .$ Then, the
coordinates of the vertices are
$A (1 + 1 - 0, 0 + 1 - 1), B (0 + 1 - 1, 1 + 0 - 1), C (0 + 1 - 1, 1 + 1 - 0)$
$i.e. A (2, 0), B (0, 0), C (0, 2)$
Hence. the coordinates of the incentre are
$(\frac{2 \times 2}{2 + 2 + \sqrt{2^{2} + 2^{2}}}, \frac{2 \times 2}{2 + 2 + \sqrt{2^{2} + 2^{2}}}) = (2 - \sqrt{2}, 2 - \sqrt{2})$

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