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

Question

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

James Atkins · Accepted Answer

#color(blue)("Radius" ~~0.5401815134)#

From diagram:

#OD=OE=OF="radius of circle"#

#Delta AOB, Delta AOC, Delta BOC# all have bases that are the sides of #Delta ABC#.

They also all have heights #r# ( the radius of the circle.)

So it follows:

#"Area of " DeltaABC= "Area of "Delta AOB+ Delta AOC+ Delta BOC#

#"Area of " DeltaABC= "1/2AB*r+ 1/2AC*r+ 1/2BC*r#

#"Area of " DeltaABC= "1/2*r(AB+ AC+ BC)#

#"Area of " DeltaABC= "1/2*rxx#perimeter of ABC

Let:

#A=(5,2)# , #B=(2,3)#, #C=(3,4)#

Using the distance formula:

#|AB|=sqrt((2-5)^2+(3-2)^2)=sqrt(10)#

#|BC|=sqrt((3-2)^2+(4-3)^2)=sqrt(2)#

#|AC|=sqrt((3-5)^2+(4-2)^2)=sqrt(8)=2sqrt(2)#

Now we need to find area of #Delta ABC#.

There are different ways we can do this. Having the lengths of the sides we could use Heron's formula, or we could find the height of ABC using line equations. Since we have surds for the lengths of the sides, Heron's formula will be pretty messy and a calculator or computer will be really helpful.

Heron's formula is given as:

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

Where #a,b,c# are the sides of ABC and s is the semi-perimeter.

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

I won't include the calculation for this, I will just give the result.

#"Area"=2#

So to find radius:

#2=1/2xxrxx(sqrt(10)+2sqrt(2)+sqrt(2))#

#r=4/(sqrt(10)+2sqrt(2)+sqrt(2))=(2sqrt(2))/(sqrt(5)+3)~~0.5401815134#

PLOT:

Caroline Aucoin · Answer

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

radius = (Area of the triangle) / (Semiperimeter of the triangle)

First, we need to calculate the area of the triangle using the coordinates given. We can use the formula for the area of a triangle given its vertices:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Then, we calculate the semiperimeter of the triangle, which is the sum of the lengths of its three sides divided by 2:

Semiperimeter = (side1 + side2 + side3) / 2

Finally, we can use the formula for the radius of the inscribed circle:

radius = Area / Semiperimeter

By substituting the coordinates of the vertices into the formulas, we can calculate the radius of the inscribed circle.