Question

AB is a line segment and M is its mid point. Semicircle are drawn with AM, MB and AB as diameter on the same side of aa line AB. A circle C(0,r)is drawn so that it touches all three semi circle. Prove that r=1/6AB​

Answer 1

r = AB/6 if a circle touches three semicircles drawn with AM, MB and AB as diameter on the same side of a line AB

Step-by-step explanation:

Let say AB = 4x

Then Radius of bigger SemiCircle = 4x/2 = 2x

AM = BM = 4x/2 = 2x

Then Radius of Smaller Semicircle = 2x/2 = x

let say r is the radius od circle touching all three semicircles at P , Q & R

OL = x + r

ON = x + r

LN = 2x

Δ OLN is an isoscelles triangle

M is midpoint hence

OM ⊥ LN

OM = 2x - r

ON² = OM² + MN²

=>(x + r)² =  (2x - r)² + x²

=> x² + r² + 2xr = 4x² + r² - 4xr + x²

=> 4x² - 6xr = 0

=> 4x = 6r

=> r = 4x/6

4x =AB

=> r = AB/6

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Answer 2

Step-by-step explanation:

here you go thank you