A triangle has corners at #(4 , 5 )#, #(3 ,2 )#, and #(5 ,3 )#. What is the radius of the triangle's inscribed circle?

1 Answer

The radius #r=0.654929" "#unit

Explanation:

We use the formula for radius of inscribed circle in a triangle

#r=sqrt(((s-a)(s-b)(s-c))/s)" "# where a,b,c are the sides of the triangle and #s# is half of the Perimeter's measure.

#s=(a+b+c)/2#

Using distance formula we compute for the sides

Let side #a# the distance from #(4, 5)# to #(3, 2)#
Let side #b# the distance from #(5, 3)# to #(3, 2)#
Let side #c# the distance from #(4, 5)# to #(5, 3)#

#a=sqrt10#
#b=sqrt5#
#c=sqrt5#

#s=(a+b+c)/2=(sqrt10+sqrt5+sqrt5)/2=(sqrt10+2sqrt5)/2#

Compute #r#

#r=sqrt(((s-a)(s-b)(s-c))/s)#

#r=sqrt((((sqrt10+2sqrt5)/2-sqrt10)((sqrt10+2sqrt5)/2-sqrt5)((sqrt10+2sqrt5)/2-sqrt5))/((sqrt10+2sqrt5)/2))#

#r=0.654929" "#unit

desmos

God bless....I hope the explanation is useful.