A triangle has corners at $(2 ,3 )$ , $(1 ,2 )$ , and $(5 ,8 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-03-02T12:50:10

$radiusapprox1.8$ units

Let the vertices of $DeltaABC$ are $A(2,3)$ , $B(1,2)$ and $C(5,8)$ .

Using distance formula,

$a=BC=sqrt((5-1)^2+(8-2)^2)=sqrt(2^2*13)=2*sqrt(13)$

$b=CA=sqrt((5-2)^2+(8-3)^2)=sqrt(34)$

$c=AB=sqrt((1-2)^2+(2-3)^2)=sqrt(2)$

Now, Area of $DeltaABC=1/2|(x_1,y_1,1), (x_2,y_2,1),(x_3,y_3,1)|$

$=1/2|(2,3,1), (1,2,1),(5,8,1)|=1/2|2*(2-8)+3*(1-5)+1*(8-10)|=1/2|-12-12-2|=13$ sq. units

Also, $s=(a+b+c)/2=(2*sqrt(13)+sqrt(34)+sqrt(2))/2=approx7.23$ units

Now, let $r$ be the radius of triangle's incircle and $Delta$ be the area of triangle, then

$rarrr=Delta/s=13/7.23approx1.8$ units.

A triangle has corners at (2 ,3 )(2 ,3 ), (1 ,2 )(1 ,2 ), and (5 ,8 )(5 ,8 ). What is the radius of the triangle's inscribed circle?