A triangle has corners at #(6 , 3 )#, #(1 ,5 )#, and #(2 ,5 )#. What is the radius of the triangle's inscribed circle?

Question

A triangle has corners at #(6  ,  3      )#, #(1   ,5  )#, and #(2  ,5   )#.  What is the radius of the triangle's inscribed circle?

Violet Aldrich · Accepted Answer

#color(indigo)("Radius of incircle " r = A_t / s = 1.55 / 5.92 ~~ 0.26 " units"#

#"Incircle radius " r = A_t / s#

#A(6,2), B(1,5), C(2,5)#

#a = sqrt((1-2)^2 + (5-5)^2) = 1#

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

#c = sqrt((6-1)^2 + (2-5)^2) = 5.83#

#"Semi-perimeter " s = (a + b + c) / 2 = (1 + 5 + 5.83) / 2 = 5.92#

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

#A_t = sqrt(5.92 (5.92-1) (5.92 - 5) (5.92 - 5.83)) = 1.55#

#color(indigo)("Radius of incircle " r = A_t / s = 1.55 / 5.92 ~~ 0.26 " units"#

Parker Allen · Answer

undefined The radius of the inscribed circle of a triangle can be calculated using the formula:

$$ r = \frac{{2 \cdot \text{{Area}}}}{{\text{{Perimeter}}}} $$

where the area of the triangle is given by Heron's formula:

$$ \text{{Area}} = \sqrt{s(s-a)(s-b)(s-c)} $$

and the perimeter ($P$) and semi-perimeter ($s$) are calculated as:

$$ P = a + b + c $$
$$ s = \frac{{a + b + c}}{2} $$

The lengths of the sides of the triangle ($a$, $b$, and $c$) can be found using the distance formula between the given points. Once you have the lengths of the sides, you can substitute them into the formulas to find the radius of the inscribed circle.