A triangle has corners at $(5, 1)$ $(5 ,1 )$ , $(7, 6)$ $(7 ,6 )$ , and $(2, 3)$ $(2 ,3 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-07-05T07:31:48

The radius of the incircle is $= 1.28$ $=1.28$

The area of the triangle is

$A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|$

$A=1/2|(5,1,1),(7,6,1),(2,3,1)|$

$=1/2(5*|(6,1),(3,1)|-1*|(7,1),(2,1)|+1*|(7,6),(2,3)|)$

$=1/2(5(6-3)-1(7-2)+1(21-12))$

$=1/2(15-5+9)$

$=1/2|19|=19/2$

The length of the sides of the triangle are

$a=sqrt((7-5)^2+(6-1)^2)=sqrt(29)$

$b=sqrt((7-2)^2+(6-3)^2)=sqrt34$

$c=sqrt((5-2)^2+(1-3)^2)=sqrt13$

Let the radius of the incircle be $=r$

Then,

The area of the circle is

$A=1/2r(a+b+c)$

The radius of the incircle is

$r=(2a)/(a+b+c)$

$=(19)/(sqrt29+sqrt34+sqrt13)$

$=19/14.82=1.28$

A triangle has corners at (5 ,1 )(5,1)(5 ,1 ), (7 ,6 )(7,6)(7 ,6 ), and (2 ,3 )(2,3)(2 ,3 ). What is the radius of the triangle's inscribed circle?