A triangle has corners at #(3 , 4 )#, #(5 ,6 )#, and #(2 ,1 )#. What is the radius of the triangle's inscribed circle?

Question

A triangle has corners at #(3  , 4    )#, #(5   ,6  )#, and #(2 ,1   )#.  What is the radius of the triangle's inscribed circle?

Wyatt Anderson · Accepted Answer

The radius is #=0.34#

The area of the triangle is

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

#A=1/2|(3,4,1),(5,6,1),(2,1,1)|#

#=1/2(3*|(6,1),(1,1)|-4*|(5,1),(2,1)|+1*|(5,6),(2,1)|)#

#=1/2(3(6-1)-4(5-2)+1(5-12))#

#=1/2(15-12-7)#

#=1/2|-4|=2#

The length of the sides of the triangle are

#a=sqrt((5-3)^2+(6-4^2))=sqrt(8)#

#b=sqrt((5-2)^2+(6-1)^2)=sqrt34#

#c=sqrt((3-2)^2+(4-1)^2)=sqrt10#

Let the radius of the incircle be #=r#

Then,

The area of the circle is

#A=1/2r(a+b+c)#

The radius of the incircle is

#r=(2a)/(a+b+c)#

#=(2*2)/(sqrt8+sqrt34+sqrt10)#

#=0.34#

Mason Adams · Answer

undefined The radius of the triangle's inscribed circle can be found using the formula:

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

Where the area of the triangle can be calculated using Heron's formula, and the perimeter is the sum of the lengths of the triangle's sides.

Given the coordinates of the triangle's vertices: (3, 4), (5, 6), and (2, 1), we can calculate the lengths of the sides using the distance formula between two points. Then, we can find the semi-perimeter, area, and ultimately the radius of the inscribed circle.