A triangle has corners at #(5 , 2 )#, #(4 ,7 )#, and #(5 ,6 )#. What is the radius of the triangle's inscribed circle?

1 Answer
sankarankalyanam
Jun 20, 2018

#color(green)("Radius of incircle " r = A_t / s = 0.4 " units"#

Explanation:

![http://mathibayon.blogspot.com/2015/01/http://derivation-of-formula-for-radius-of-incircle.html](https://useruploads.socratic.org/tJxrUQbRPS2AxwcUMf6J_incircle%20radius.png)

#"Incircle radius " r = A_t / s#

#A(5,2), B(4,7), C(5,6)#

#a = sqrt((4-5)^2 + (7-6)^2) = sqrt2#

#b = sqrt((5-5)^2 + (6-2)^2) = 4#

#c = sqrt((5-4)^2 + ( 2-7)^2) = sqrt26#

#"Semi-perimeter " s = (a + b + c) / 2 = (sqrt2 + 4 + sqrt26) / 2 = 5.27#

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

#A_t = sqrt(5.27 (5.27-sqrt2) (5.27 - 4) (5.27 - sqrt26)) =2.1 #

#color(green)("Radius of incircle " r = A_t / s = 2.1 / 5.27 = 0.4 " units"#

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