The incenter of the triangle formed by the points $(0,0,0),(3,0,0)$ and $(0,4,0)$ is

Question

Aicharya · Accepted Answer

Let $A(0,0,0), B(3,0,0), C(0,4,0)$

Now, $ \begin{aligned} & a=\sqrt{(0-3)^2+(4-0)^2+(0-0)^2}=B C \\ & a=\sqrt{25}=5 \end{aligned} $ and $b=A C=\sqrt{0+4^2+0}=4$ and $c=A B=\sqrt{3^2+0+0}=3$ Now, coordinate of incentre $(x, y, z)$ is given as $ x=\frac{a x_1+b x_2+c x_3}{a+b+c} $ $ \begin{aligned} & =\frac{15(0)+(4)(3)+(3)(0)}{5+4+3}=1 \\ y & =\frac{a y_1+b y_2+c y_3}{a+b+c}=\frac{(5)(0)+(4)(0)+(3)(4)}{5+4+3}=1 \\ z & =\frac{a z_1+b z_2+c z_3}{a+b+c} \\ & =\frac{(5)(0)+(4)(0)+(3)(0)}{5+4+3}=0 \\ \therefore(x, y, z) & =(1,1,0) \end{aligned} $

The incenter of the triangle formed by the points $(0,0,0),(3,0,0)$ and $(0,4,0)$ is

Solution