Given a rectangle ABCD: AB=6, BC (the larger side) not disclosed. In this rectangle 3 circles are inscribed such that the biggest
is touching sides BC & CD and AD, the smallest of (diameter =3 ) touches sides AB & BC and between those two a 3rd one (diameter =4) is touching both two others and the side AD.
Evaluate the area of the rectangle.
area =71.0908
Call the circles C3, C4 and C6. Put A at the origin
with B,C,D going around clockwise. C3 is snug in the
top left corner with center (1/5, 4.5). C6 is boxed-in at the right.
C4 is on the floor at (x,2). The straight line connecting C3 and C4
is 1.5 + 2 = 3.5 units. Using the two centers to give this distance:
(x-1.5)^2 + (4.5 -2)^2 = 3.5 ^2
gives x = (3/2) + sqrt(6) = 3.969
Likewise, the straight line between C4 and C6 is 5 units and
from the two centers: delta_X^2 + 1^2 = 5^2
delta_X = 2 sqrt(6) = 4.9000, where delta_X is the
x-distance between the two centers C4 and C6.
AD = x + delta_X + 3 = (3/2) + sqrt(6) + 2 sqrt(6) + 3
= 3 sqrt(6) + 9/2 = 11.848
Area = 6 * AD = 18 sqrt(6) + 27 = 71.0908
Edited on March 17, 2023, 1:09 am
soln
