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

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Answer 1 · 2016-06-15T16:24:47

$r= 0.716$

Given the coordinates of three vertices of $DeltaABC$ are

$"Corodinate of A " x_A=5,y_A=8$

$"Corodinate of B " x_B=2,y_B=8$

$"Corodinate of C " x_C=3,y_C=1$

Lengths of three sides

$AB=sqrt((x_A-x_B)^2+(y_A-y_B)^2)$

$=sqrt((5-2)^2+(8-3)^2)=sqrt34$

$BC=sqrt((x_B-x_C)^2+(y_B-y_C)^2)$

$=sqrt((2-3)^2+(3-1)^2)=sqrt5$

$CA=sqrt((x_C-x_A)^2+(y_C-y_A)^2)$

$=sqrt((3-5)^2+(1-8)^2)=sqrt53$

Now area of $DeltaABC$

$=|1/2(y_A(x_B-x_C)+y_B(x_C-x_A)+y_C(x_A-x_B))|$

$=|1/2(8(2-3)+3(3-5)+1(5-2))|$

$=|1/2(-8-6+3)|=5.5$

If the radius of the incircle of $DeltaABC " "be" "r$ then

$1/2(AB+BC+CA)xxr="area "DeltaABC$

$=>1/2(sqrt34+sqrt5+sqrt53)*r=5.5$

$=>r=(2xx5.5)/(sqrt34+sqrt5+sqrt53)$

$:.r= 0.716$

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