First slide

Cartesian plane

Question

If (0, 1), (1, 1) and (1, 0) are the mid points of the sides of a triangle, the coordinates of its incentre are

Moderate

A

$(2 + \sqrt{2}, 2 + \sqrt{2})$
B

$((2 + \sqrt{2}), - (2 + \sqrt{2}))$
C

$((2 - \sqrt{2}), (2 - \sqrt{2}))$
D

$((2 - \sqrt{2}), - (2 - \sqrt{2}))$

Solution

The given triangle is right-angled with vertices
$A (0, 0), B (2, 0) a n d C (0, 2) .$
Coordinates of the incentre are
$\begin{array}{l} (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c}) \\ = (\frac{2 \sqrt{2} \times 0 + 2 \times 2 + 2 \times 0}{2 \sqrt{2} + 2 + 2}, \frac{2 \sqrt{2} \times 0 + 2 \times 0 + 2 \times 2}{2 \sqrt{2} + 2 + 2}) \\ = (2 - \sqrt{2}, 2 - \sqrt{2}) \end{array}$

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