Question

The coordinates of the in centre of the triangle whose vertices are (0, 0), (4, 0) and (0, 3) are __________.

Answer 1

In a triangle with vertices (0, 0), (4, 0) and (0, 3)

Since co-ordinate of the in-centre of the triangle ABC are
$\left(\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c},\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c}\right)\mathrm{where}a=BCb=ACc=AB$
$$\begin{gathered} \mathrm{Here}a=BC=\sqrt{{\left(4-0\right)}^2+{\left(0-0\right)}^2}=4 \\ b=AC=\sqrt{4^2+9}=\sqrt{25}=5 \\ c=AB=\sqrt{0+9}=3 \end{gathered}$$
∴ Co-ordinates of in-centre of the triangle ABC
$$\begin{gathered} =\left\{\left(\frac{4\times 0+5\times 0+3\times 4}{12}\right),\left(\frac{4\times 3+5\times 0+3\times 0}{12}\right)\right\} \\ =\left\{\left(\frac{12}{12},\frac{12}{12}\right)\right\} \end{gathered}$$
i.e co-ordinates of in-centre = (1, 1)