Semi-semicircle

Geometry Level 4

Take a semicircle of radius R R and one semicircle k k times bigger and make them tangent like in the picture above. Then construct a 90 ∘ {{90}^{\circ }} arc through the midpoints of these 2 semicircles. If the shaded area is A A , calculate the ratio A k R 2 \dfrac{A}{kR^2} .

Hint : Notice that all three centers are collinear.


The answer is 1.

Adrian Aruștei by Adrian Aruștei , 27, Romania

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1 solution

Ahmad Saad
Jun 2, 2016

21 Helpful 0 Interesting 0 Brilliant 0 Confused

sir Ahmed exllent solution
yours aly mira (almasry ) ihope you well

Aly Ahmed - 5 years ago

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Hello, Mr Aly I missed you.

Ahmad Saad - 5 years ago

Exllent solution,short and sweet solution. With best wishes for your health.

Niranjan Khanderia - 4 years, 4 months ago

Can anyone explain what is being done in 2nd step,also how does r dash intersect at that point

Tarun B - 3 years, 8 months ago

Perfect solution

محمد مصباح - 5 years ago

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