Question

AB is a line segment of length 24 cm and C is its middle
point. On AB, AC and CB semi-circles are described.
Determine the radius of the circle which touches all the
three semi-circles.​

Answer 1

Answer:

C is its midpoint. On AB, AC and BC semicircles are described. Hence, radius of the circle which touches all three semicircles is 4 cm.

Step-by-step explanation:

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.

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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