If You Solve This, Your Brain Is Special
4 Video Views·Nov 28, 2025
Dive into a fascinating Math Olympiad geometry problem! In this video, we tackle a challenging geometry puzzle that requires a clever approach. Watch as we break down the problem step-by-step and reveal a surprisingly elegant solution. Perfect for anyone looking to sharpen their geometry skills or prepare for math competitions
In this video, we have to find shaded area, so both the semicircles are congruent.
That means area of semicircle AB, it will be equal to area of semicircle BC.
And suppose this area is A, this area is B, and this area is C, then A plus B, it will be B plus C,
and B will get cancelled, so A, it will be C, and C, it is shaded area, and suppose this point
on AB, suppose this point is Q, and join AP, suppose PQ is H, and in any circle diameter
makes an angle of 90 degrees on the circle, so angle APB, it will be 90 degrees, and area
of ABP, it will be equal to area of ABP, and area of ABP, it is 1 over 2 times base is AB,
and height is PQ, and area of ABP, it will be 1 over 2 times base is AP, and height is BP,
and 1 over 2 will get cancelled, and AB is 5, and PQ is H, and AP, it will be 5 times
sin 15 degrees, and BP, it will be 5 cos 15 degrees, and 5 will get cancelled, so H,
it is 5 times sin 15 degrees times cos 15 degrees, and sin A cos A, it is sin 2 A over 2,
so H, it is 5 times sin 15 degrees cos 15 degrees, it will be sin 30 degrees over 2,
and H, it is 5 times 1 over 2 times sin 30 degrees is 1 over 2, that will be 5 over 4,
so H, it is 5 over 4, and now in any circle, any chord, suppose AP is making an angle theta
on the circle, suppose this point is B, then at the center O, it will make an angle of 2 theta,
and suppose the center of the circle is O, and if we join PO, then angle AOP, it will be 2 times
angle ABP, that will be 2 times 15 degrees, and it is 30 degrees, this angle is 30 degrees,
and area of sector AOP, it will be 5 times radius square times angle is 30 degrees over
360 degrees plus area of triangle BOP, it will be 1 over 2 times base is BO, and height is PQ,
and radius, it will be AO, that will be BO, and it will be AB over 2, and AB is 5 over 2,
so BO is 5 over 2, it is pi times 5 over 2 A square times 1 over 2 N plus 1 over 2 times BO
is 5 over 2, and PQ is 5 over 4, and it is 25 pi over 48 plus 25 over 16, and it is 25 over 16
times pi over 3 plus 1, it is 25 over 16 times pi over 3 plus 1.