Question

AB is a line segment of length 48 cm, C is its middle point. On AB, AC, CB semicircles are described. The radius of the circle inscribed in the space enclosed by three semicircles.​

Answer 1

the radius is 8 cm

• AB = 48 cm, AC = BC = 24 cm, AX = CX = CY = YB = 12 cm

• Let the radius be r cm

• OX = radius of smaller semicircle + r

• CX = radius of smaller semicircle = 12 cm

• In triangle YOC, OX^2 = CX^2 + OC^2   =>  OC^2 = OX^2 - CX^2 = (12+r)^2 - 12^2 = r^2 + 24r  ⇒  OC = (r^2 + 24r)^0.5
• DC = DO + OC  ⇒  24 = r +(r^2 + 24r)^0.5  ⇒  24 - r = (r^2 + 24r)^0.5        ⇒  (24 - r)^2  =  (r^2 + 24r)  ⇒  576 - 48r + r^2 = r^2 + 24r  ⇒  72r = 576   ⇒  r = 8 cm