The diagram above shows 3 identical circles placed inside a right tria

The above diagram shows 3 identical circles placed inside a right triangle so that all 3 circles are tangent to line AB. What is the radius of each circle?

A) 5
B) 6
C) 7
D) 8
E) 10

Solution

• Let the radius of each circle be r.
• It is given that AB is tangent to all three identical circles, So, radius of each circle will be perpendicular to AB at the point of tangency.

o So, DE is perpendicular to AB.
o Also, \(AG = DE = r\)

 Thus, as shown in the below diagram, AEDG is a rectangle,


• And, \(AE = GD = 5r\)
• Also, \(∠CGD = 90\)°, and triangle CGD is a right-angled triangle.
• Now, using Pythagoras theorem in triangle, CGD, we get:

o \(CD^2 = CG^2 + GD^2 ⟹ CD^2 = (30-r)^2 + (5r)^2 …………Eq.(i)\)
• Now, in the right-angled triangles DEB and DFB,

o Side \(DE = DF = r\) and

o hypotenuse is common, i.e. DB
o So, EB must be equal to FB i.e. \(EB = FB = 40 – 5r\)
• So, \(FC = BC -BF = 50 – (40 – 5r) = 10+ 5r \)
• Now, using Pythagoras theorem in triangle in triangle, CDF,

o \(CD^2 =DF^2 +FC^2 ⟹ CD^2 = (10+5r)^2 + (r)^2 ……………….Eq.(ii)\)
• From Eq.(i) and (ii), we can write,

o \((30-r)^2 + (5r)^2 = (10+5r)^2 + (r)^2\)
o Or, \(900 + r^2 – 60r + 25r^2 = 100 + 25r^2 + 100r + r^2\)
o Or, \(160r = 800 ⟹ r = 5\)