Semicircles

Geometry Level 3

AB is a line segment and M is its midpoint. Semicircles are drawn with AB , AM and MB as diameters on the same side of line AB. A circle with center O is drawn which touches the three semicircles, then the radius of the smallest cicle is equal to x A B y \frac{ x AB }{y} .

Find x + y x + y .

This problem is a part of the sets - 1's & 2's & " G " for geometry .


The answer is 7.

Sakanksha Deo by Sakanksha Deo , 23, India

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2 solutions

Consider right △ C O M \triangle COM , by pythagorean theorem we have

( r + x ) 2 = r 2 + ( 2 r − x ) 2 (r+x)^2=r^2+(2r-x)^2

r 2 + 2 r x + x 2 = r 2 + 4 r 2 − 4 r x + x 2 r^2+2rx+x^2=r^2+4r^2-4rx+x^2

2 r x = 4 r 2 − 4 r x 2rx=4r^2-4rx

6 r x = 4 r 2 6rx=4r^2

6 x = 4 r 6x=4r

x = 4 r 6 = A B 6 x=\dfrac{4r}{6}=\dfrac{AB}{6}

Therefore, the desired answer is 1 + 6 = 7 1+6=\boxed{7}

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Ahmad Saad
Nov 11, 2015

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