## Wednesday, April 27, 2011

### Rectangle, Circles inscribed

In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.

UPDATE!

Triangle Formulas : Right Triangle
C = A + B = Pi/2 radians = 90 degrees
c^2 = a^2 + b^2
P = a + b + c
s = (a+b+c)/2
K = ab/2

Particular case: Circle Inscribed in Triangle 3, 4, 5

Circle Inscribed in a Right Triangle

General case: Circle Inscribed in Triangle a, b, c
We can write,
r = ab/(a + b + c) with c = sqrt(a^2 + b^2)
d = sqrt((a - 2r)^2+(b - 2r)^2)

OR

r = (a + b - c)/2 where c = Sqrt(a^2 + b^2)
Then (a - 2r)^2 + (b - 2r)^2 = (c - a)^2 + (c - b)^2 etc

#### 1 comment:

1. Two ways to solve this one, first, find the radius of the inscribed circle using
Sqrt[((s - a) (s - b) (s - c))/s] where s=semi perimeter and sides 5,12,13 gives a radius of 2. So centre of first circle would be 2,2, and we can work out the centre of the second to be 10,3. The distance between centres would then be.

Sqrt(65).

We could also use the cartesian co-ordinates equation of the co-ordinates of the incircle

xi,yi =
a*x1+b*x2+c*x3/a+b+c, a*y1+b*y2+c*y3/a+b+c

substituting the appropriate sides and co-ordiantes will give us centres at 2,2 and 10,3.

We would get 60/30,60/30 and 300/30, 90/30 using the co-ordinates 0,0, 12,0, 12,5 and 0,5.

Paul