A triangle has corners at $(4, 5)$ $(4 , 5 )$ , $(1, 3)$ $(1 ,3 )$ , and $(5, 3)$ $(5 ,3 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-07-21T07:21:53

The radius of the incircle is $= 0.81 u$ $=0.81u$

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The corners are

$A = (4, 5)$ $A=(4,5)$

$B = (1, 3)$ $B=(1,3)$

$C = (5, 3)$ $C=(5,3)$

The length of the sides of the triangle are

$c = \sqrt{{(1 - 4)}^{2} + {(3 - 5)}^{2}} = \sqrt{9 + 4} = \sqrt{13} = 3.61$ $c=sqrt((1-4)^2+(3-5)^2)=sqrt(9+4)=sqrt13=3.61$

$a = \sqrt{{(5 - 1)}^{2} + {(3 - 3)}^{2}} = \sqrt{16 + 0} = \sqrt{14} = 4$ $a=sqrt((5-1)^2+(3-3)^2)=sqrt(16+0)=sqrt14=4$

$b = \sqrt{{(5 - 4)}^{2} + {(3 - 5)}^{2}} = \sqrt{1 + 4} = \sqrt{5} = 2.24$ $b=sqrt((5-4)^2+(3-5)^2)=sqrt(1+4)=sqrt5=2.24$

The area of the triangle is

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

$=1/2(x_1(y_2-y_3)-y_1(x_2-x_3)+(x_2y_3-x_3y_2))$

$A=1/2|(4,5,1),(1,3,1),(5,3,1)|$

$=1/2(4*|(3,1),(3,1)|-5*|(1,1),(5,1)|+1*|(1,3),(5,3)|)$

$=1/2(4(3-3)-4(1-5)+1(3-15))$

$=1/2(0+20-12)$

$=1/2|8|=4$

The radius of the incircle is $=r$

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

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

$=8/(9.85)=0.81$

A triangle has corners at (4 , 5 )(4,5)(4 , 5 ), (1 ,3 )(1,3)(1 ,3 ), and (5 ,3 )(5,3)(5 ,3 ). What is the radius of the triangle's inscribed circle?