A triangle has corners at #(3 , 5 )#, #(4 ,7 )#, and #(4 ,6 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-05-18T13:52:38

Radius of inscribed circle #color(brown )(r = 0.215# units

#A(3,5), B(4,7), C(4,6)#

Using distance formula,

#bar(AB) = c = sqrt((3-4)^2+(5-7)^2) = sqrt5 = 2.236#

#bar(BC) = a = sqrt((4-4)^2 + (7-6)^2) = 1#

#bar(AC) = b = sqrt((3-4)^2 + (5-6)^2) = sqrt2 = 1.414#

Area of triangle knowing all three sides is given by

#A_t = sqrt(s (s-a)(s-b)(s-c))#

Where semi perimeter = #s = (a + b + c) / 2#

#s = (1 + 1.414 + 2.236) / 2 = 4.65/2 = 2.325#

#A_t = sqrt(2.325 (2.325-1) * (2.325-1.414) * (2.325 -2.236)) = 0.5#

Let r be the radius of incircle.

Then #r = A_t / s = 0.5 / 2.325 = color(brown)(0.215# units