Question

The lengths of the legs of a right triangle are 3 and 4. Two congruent circles, externally tangent to each other, are drawn inside the triangle, with each circle tangent to the hypotenuse and one of the legs. What is the distance between the centers of the two circles?

Could anyone please help me with this problem? I just can't figure it out?

Answer 1

not drawn to scale

Answer 2

Yes, ok i got that, but I have no idea what to do from there

Answer 3

i split the triangle into 6 sections, 5 smaller triangles and 1 rectangle
two tangent lines to same circle are always equal length

To solve for "r" you need to equate the areas
obviously the area of the triangle is 6, now just find area of the smaller sections in terms of r

Equations:
a+b = 4
b+c =3
a +c+ 2r = 5

by substitution:
a = 3-r
b = 1+r
c = 2 - r

Areas:
rectangle --> 2r*r = 2r^2
upper small triangle --> cr/2 = r(2-r)/2
lower small triangle --> ar/2 = r(3-r)/2
left triangle --> 3r/2
bottom triangle --> 4r/2 = 2r
center triangle ---> (2r)*b/2 = r*b = r(1+r)

add those up and set equal to 6
combine like terms and solve the quadratic using the quadratic formula

you should end up with:
\[r^2 +5r-4 = 0\]
\[r = \frac{\sqrt{41} -5}{2}\]
so distance between centers is 2r
\[2r = \sqrt{41} - 5\]