Question

Write the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12).

Answer 1

Let A(0,0), B(5, 0) and C(0, 12) be the vertices of the given triangle.
In-centre I of a triangle with vertices $A\left(x_1,y_1\right),B\left(x_2,y_2\right)\mathrm{and}C\left(x_3,y_3\right)$ is given by:

$I\equiv \left(\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c},\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c}\right)$, where a = BC, b = AC and c = AB.

Now,

$$\begin{gathered} a=BC=\sqrt{{\left(5-0\right)}^2+{\left(0-12\right)}^2}=13 \\ b=AC=\sqrt{0+{12}^2}=12 \\ c=AB=\sqrt{0+5^2}=5 \end{gathered}$$

$$\begin{gathered} \therefore I\equiv \left(\frac{13\times 0+12\times 5+5\times 0}{13+12+5},\frac{13\times 0+12\times 0+5\times 12}{13+12+5}\right) \\ \Rightarrow I\equiv \left(\frac{60}{30},\frac{60}{30}\right)=\left(2,2\right) \end{gathered}$$

Hence, the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12) is (2, 2).