A right triangle has its legs parallel to the x and y axes as shown in the figure. If the hypotenuse has a slope of -4/3, and the diameter of the bigger circle is 2020, what is the diameter of the smaller circle?
Let the center of the larger circle be (10,10)....and this is the incenter of the large triangle
Let A be the angle at the bottom right vertex
The tangent of this angle is 4/3
But the bisector of angle A creates an angle with 1/2 the measure of A
And the tangent of this angle is sqrt [ 1 -cos A] / sqrt [1 + cos A]
And cos A = 3/5
So tan (A/2) = sqrt [ 1-3/5 ] / sqrt [1 + 3/5] = 1/2
And
tan (A/2) =10/20
cos (A/2) = 2/sqrt (5)
Therefore, the distance from the bottom right vertex of the large triangle to the center of the larger circle can be found as
cos (A/2) = 20 / D
D = 20 / (2 /sqrt (5) ) = 10sqrt (5)
And using reflexive symmetry of similar polygons, the angle bisector will go through the center of both circles....
So using similar triangles we have that
10/ (10sqrt (5) ) = r / [ 10sqrt (5) - 10 - r ]
1/sqrt (5) = r / [10sqrt (5) -10 - r ] =
r sqrt (5) = 10sqrt (5) - 10 - r
r ( sqrt (5) + 1 ) = 10 ( sqrt (5) - 1)
r = 10 ( sqrt (5) -1) ( sqrt (5) -1) / 4
r = 10 ( 5 - 2sqrt (5) + 1) /4
r = 15 - 5sqrt (5)
Here's a pic :