GMATPrepNow
The diagram above shows 3 identical circles placed inside a right triangle so that all 3 circles are tangent to line AB. What is the greatest possible radius of each circle?
A) 20/3
B) 80/11
C) 22/3
D) 81/11
E) 10
Attachment:
The attachment rnMKbFE.png is no longer available
Referring to the diagram below:
Attachment:
111.JPG [ 42.47 KiB | Viewed 4757 times ]
RQ is drawn tangent to the 3rd circle as shown
Assuming each circle to be of radius r, we have: AR = 4r = PQ
=> RB = 40 - 4r
Triangles CPQ and CAB are similar
=> CP/CA = PQ/AB = CQ/CB
=> CP/30 = 4r/40 = CQ/50
=> CP = 3r and CQ = 5r
=> BQ = 50 - 5r; RB = 40 - 4r; AP = RQ = 30 - 3r
From O (center of the 3rd circle), the radii OT and OU are joined as shown
RT = OU = OT = RU = r
=> UB = RB - RU = 40 - 4r - r = 40 - 5r
=> BS = BU = 40 - 5r (tangents from the same point to a circle are equal)
=> QS = BQ - BS = (50 - 5r) - (40 - 5r) = 10
=> QT = QS = 10 (tangents from the same point to a circle are equal)
Thus:
QR = QT + TR
=> 30 - 3r = 10 + r
=> r = 5A couple of questions:
1. Why ask for the GREATEST possible radius, when the exact radius can be solved for?
2. Does the question ask for the diameter?