A triangle has corners at #(2 , 8 )#, #(5 ,7 )#, and #(3 ,1 )#. What is the radius of the triangle's inscribed circle?

Question

A triangle has corners at #(2  ,  8    )#, #(5   ,7  )#, and #(3  ,1   )#.  What is the radius of the triangle's inscribed circle?

Avery Allen · Accepted Answer

midpoint of #(2,8)# and #(5,7)# = #(3.5,7.5)#
slope of line between #(2,8)# and #(5,7)# = #(8-7)/(2-5) = -1/3#
#:.# slope of perpendicular line # = - 1/(-1/3) = 3#
point slope equation:
#y-7.5 = 3(x-3.5)#
#y = 3x-10.5+7.5#
#y = 3x-3# midpoint of #(2,8)# and #(3,1)# = #(2.5,4.5)#
slope of line between #(2,8)# and #(3,1)# = #(8-1)/(2-3) = -7#
#:.# slope of perpendicular line # = - 1/(-7) = 1/7#
point slope equation:
#y-4.5= 1/7(x-2.5)#
#y = 1/7x-5/14+4.5#
#y = 1/7x-34/7# Now we get the system of equations: 
#y = 3x-3#
#y = 1/7x-34/7# Solving, we get #(-13/20,-99/20)# The distance between #(-13/20,-99/20)# and #(3,1)# gives the radius of the circle. #=sqrt(19490)/40 ~~ 6.98#

Claire Akin · Answer

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

$$ r = \frac{2 \times \text{Area of the Triangle}}{\text{Perimeter of the Triangle}} $$

To calculate the area of the triangle, you can use the formula for the area of a triangle given its vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $:

$$ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $$

And the perimeter of the triangle can be calculated using the distances between its vertices.

After calculating the area and the perimeter, plug them into the formula to find the radius of the inscribed circle.