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.
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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
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