A triangle has corners at #(3 , 4 )#, #(8 ,2 )#, and #(1 ,8 )#. What is the radius of the triangle's inscribed circle?

Question

A triangle has corners at #(3  ,  4    )#, #(8   ,2  )#, and #(1  ,8   )#.  What is the radius of the triangle's inscribed circle?

Sophia Alfred · Accepted Answer

The radius of inscribed circle #r=0.8387131004" "#unit

Let us label the points #A(3, 4)#, #B(8, 2)#, #C(1, 8)# Solve for the distances #a=BC# and #b=AC# and #c=AB# The radius of the inscribed circle #r# formula #r=sqrt(((s-a)(s-b)(s-c))/s)# Solve for the values of s, a, b, and c first #a=sqrt((x_b-x_c)^2+(y_b-y_c)^2)# #a=sqrt((8-1)^2+(2-8)^2)# #a=sqrt85# #b=sqrt((x_a-x_c)^2+(y_a-y_c)^2)# #b=sqrt((3-1)^2+(4-8)^2)# #b=sqrt20# #c=sqrt((x_a-x_b)^2+(y_a-y_b)^2)# #c=sqrt((3-8)^2+(4-2)^2)# #c=sqrt(29)# Half the Perimeter of the triangle #s# #s=(a+b+c)/2=(sqrt85+sqrt20+sqrt29)/2# Compute #r# #r=# #sqrt((((sqrt85+sqrt20+sqrt29)/2-a)((sqrt85+sqrt20+sqrt29)/2-b)((sqrt85+sqrt20+sqrt29)/2-c))/[0.5*(sqrt85+sqrt20+sqrt29)])# #r=0.8387131004# God bless....I hope the explanation is useful.

Aurora Ayala · Answer

undefined To find the radius $ r $ of the triangle's inscribed circle, you can use the formula:

$$ r = \frac{{\text{Area of the triangle}}}{{\text{Semiperimeter of the triangle}}} $$

First, calculate the semiperimeter $ s $ of the triangle using the formula:

$$ s = \frac{{\text{sum of all three sides}}}{{2}} $$

Then, find the area $ A $ of the triangle using Heron's formula:

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

where $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle.

Once you have both the semiperimeter and the area, you can find the radius $ r $ using the first formula mentioned.