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

Question

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

Savannah Aguilar · Accepted Answer

The radius of the incircle is #=1.06#

The length of the sides of the triangle are

#c=sqrt((1-3)^2+(7-2)^2)=sqrt(4+25)=sqrt29=5.39#

#a=sqrt((5-1)^2+(4-7)^2)=sqrt(16+9)=5#

#b=sqrt((5-3)^2+(4-2)^2)=sqrt(4+4)=sqrt8=2.83#

The area of the triangle is

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

#=1/2(x_1(y_2-y_3)-y_1(x_2-x_3)+(x_2y_3-x_3y_2))#

#A=1/2|(3,2,1),(1,7,1),(5,4,1)|#

#=1/2(3*|(7,1),(4,1)|-2*|(1,1),(5,1)|+1*|(1,7),(5,4)|)#

#=1/2(3(7-4)-2(1-5)+1(4-35))#

#=1/2(9+8-31)#

#=1/2|-14|=7#

The radius of the incircle is #=r#

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

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

#=14/(13.22)=1.06#

Wyatt Anderson · Answer

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

$$ r = \frac{{\text{Area of the triangle}}}{{\text{Semi-perimeter of the triangle}}} $$

First, find the lengths of the sides of the triangle using the distance formula:

$$ \text{Side 1} = \sqrt{(3 - 1)^2 + (2 - 7)^2} $$
$$ \text{Side 2} = \sqrt{(3 - 5)^2 + (2 - 4)^2} $$
$$ \text{Side 3} = \sqrt{(1 - 5)^2 + (7 - 4)^2} $$

Then, calculate the semi-perimeter of the triangle:

$$ s = \frac{{\text{Side 1} + \text{Side 2} + \text{Side 3}}}{2} $$

Next, compute the area of the triangle using Heron's formula:

$$ \text{Area} = \sqrt{s(s - \text{Side 1})(s - \text{Side 2})(s - \text{Side 3})} $$

Finally, plug the area and semi-perimeter values into the formula for the radius:

$$ r = \frac{{\text{Area}}}{{s}} $$

This will give you the radius of the inscribed circle of the triangle.