If sides of a triangle 3,4,5 cm .Find the distance between incentre and circumcentre.

Guest Nov 11, 2019

#1**+1 **

Triangle

Sides: a = 3 b = 4 c = 5

Area: T = 6

Perimeter: p = 12

Semiperimeter: s = 6

Inradius: r = 1

Circumradius: R = 2.5

The distance between the incenter and the circumcenter is sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula.

The Distance=Sqrt(2.5*(2.5 - 2*1))

=Sqrt(2.5*(0.5))

=Sqrt(1.25)

** =1.118 cm - distance between incenter and circumcenter.**

Guest Nov 11, 2019

#2**+1 **

Let A = (0, 0)

Let B = (0, 3)

Let C = (4, 0)

We can find the incenter thusly :

[ Length of side opposite A * xcoordinate of A + Length of side opposite B * x coordinate of B + Length of side opposite C*xcoordinate of C ] , [ Length of side opposite A * ycoordinate of A + Length of side opposite B * y coordinate of B + Length of side opposite C*ycoordinate of C ]

( [(5(0) + 0 (4) + 4(3)] / 12 , [5(0) + 3(4) + 0(3) ] / 12 ) =

(1 , 1 )

In a right triangle.....the circumcenter is located at the midpoint of the hypotenuse....this point is

(2, 3/2)

So....the distance between the incenter and the circumcenter is

sqrt [ ( 2 - 1)^2 + (3/2 - 1)^2 ] =

sqrt [ 1*2 + (1/2)^2 ] =

sqrt [ 1 + 1/4 ] =

sqrt [ 5/4] = sqrt (5) / 2

CPhill Nov 11, 2019