A triangle has corners at #(5 , 2 )#, #(1 ,3 )#, and #(7 ,4 )#. What is the radius of the triangle's inscribed circle?

1 Answer
sankarankalyanam
Jun 21, 2018

#color(blue)("Radius of incircle " r = A_t / s = 5.0224 / 6.52 = 0.77 " units"#

Explanation:

![http://mathibayon.blogspot.com/2015/01/http://derivation-of-formula-for-radius-of-incircle.html](https://useruploads.socratic.org/qByQYJn5SEeAPjtKhB4j_incircle%20radius.png)

#"Incircle radius " r = A_t / s#

#A(5,2), B(1,3), C(7,4)#

#a = sqrt((7-1)^2 + (4-3)^2) = sqrt37#

#b = sqrt((7-5)^2 + (4-2)^2) = sqrt8#

#c = sqrt((5-1)^2 + ( 2-3)^2) = sqrt17#

#"Semi-perimeter " s = (a + b + c) / 2 = (sqrt37 + sqrt8 + sqrt17) / 2 = 6.52#

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

#A_t = sqrt(6.52 (6.52-sqrt37) (6.52 - sqrt8) (6.52 - sqrt17)) =5.0224#

#color(blue)("Radius of incircle " r = A_t / s = 5.0224 / 6.52 = 0.77 " units"#

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