Semicircle Squeeze
Two small semicircles are inscribed in a larger semicircle, and a circle is drawn such that it is internally tangent to all three semicircles, as seen above. If the diameter of the biggest semicircle is 12, find the radius of the small circle.
The answer is 2.
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2 solutions
Just did it in seconds.... Note: Radius of smaller circle ( r ) = 6 1 of Diameter of bigger circle .
⇒ r = 6 1 × 1 2 = 2 .
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Yes, this works! Can you prove it?
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Yes, This can be proved in the same way as Jason did.
In right-angled △ .
Using Pythagoras.
⇒ ( 4 d + r ) 2 = ( 4 d ) 2 + ( 2 d − r ) 2
⇒ 2 r = 4 d − r
∴ r = 6 d .
In response to abhay you can not judge it like that
But how its a right angle traingle?
Big thanks
By using Pythagorean Theorem, we can conclude that
( 6 − r ) 2 + 3 2 = ( 3 + r ) 2
3 6 − 1 2 r + r 2 + 9 = 9 + 6 r + r 2
3 6 = 1 8 r
r = 2